We consider the box dimension of the graphs of the generalized Riemann-type functions G_δ(x)=∑_n=1^∞g(n^2x)n^-1-δ with 1-periodic real-valued continuous functions g and 0<δ≤ 1. Firstly, we establish a rational-level non-vanishing criterion for the lower bound of lower box dimension of the graph of G_δ. More precisely, We prove that the lower bound _B(graph G_δ)≥7/4-δ/2 under a mild decay condition of the Fourier coefficients of g and non-vanishing of the square-class chirp functional S_d(a;q) at a single rational a/q. A resolution theorem then asserts that for any nonconstant real trigonometric polynomial g, the chirp functional S(a;q) cannot vanish at every rational simultaneously; consequently, _B(graph G_δ)=7/4-δ/2 for all such g with 0<δ≤1 which gives a negative answer to . Finally, two guiding examples distinguish structural vanishing from genuinely arithmetic vanishing related to modular elliptic curve and governed by the Prime Number Theorem.

We investigate the box dimension of the graphs of a class of continuous periodic functions G_δ(x)=∑_n=1^∞g(n^2x)n^-1-δ with 1-periodic Lipschitz functions g and 0<δ≤ 1, which generalizes the result of the classical Riemann function corresponding to g(x)=sin(2πx) and δ=1. More precisely, we first prove that the lower box dimension of the graph of G_δ is no less than 7/4-2 when the Fourier coefficients of g satisfy an arithmetic non-vanishing condition related to the distribution of quadratic residues. This result is new and non-trivial even when g has a finite Fourier expansion, highlighting the intrinsic arithmetic complexity of the series. Secondly, if g' is Lipschitz continuous on ℝ, we show that the upper box dimension does not exceed 7/4-2, which extends earlier work of Chamizo and Córdoba and reveals deep connection between the regularity of g and the fractal dimension of the associated Riemann-type series. In the end, we give some illustrative examples and propose some further problems.

In this paper, we study an elliptic variational problem regarding the p$$ p $$ ‐fractional Laplacian in ℝN$$ {\mathrm{\mathbb{R}}}^N $$ on the basis of recent result which generalizes some nice published work, and then give some sufficient conditions under which some weak solutions to our studied elliptic variational problem are continuous in ℝN$$ {\mathrm{\mathbb{R}}}^N $$ . In the final appendix, we correct the proofs of two published lemmas for 1

Let f be a hyperbolic rational map with degree $$d\ge 2$$ whose Julia set is connected. We give an elementary approach to prove that there exists a rational map g with degree $$\le 7d-2$$ such that g contains a buried Julia component which is homeomorphic to the Julia set of f.

In this paper, we obtain the precise form of meromorphic functions u(z(1), z(2)) in C-2 when the linearly independent operators au(z1) + bu(z2) and cu(z1) + du(z2 )have a common right factor with constants a, b, c, d. We also describe entire solutions of the partial differential equationF(u(z1), u(z2), ... , u(zn)) = 1in C-n under a more general definition in the sense of the prime functions. In addition, our results generalize the recent results in Li [22, 27].

By using Nevanlinna theory and linear algebra, we show that the number one is a lower bound of the hyper-order of any meromorphic solution of a nonlinear differential–difference equation under certain conditions.

Let F-lambda(z) = lambda exp (z)/z with parameter lambda is an element of C. We consider the M-set consisting of all parameters lambda for which F-lambda has empty Fatou set. It is known that lambda = 1 belongs to the M-set. In this article, we show that lambda = 1 is a density point of the M-set, which means that the probability that F-lambda has empty Fatou set approaches 1 as lambda approaches 1.

In this paper, we investigate the Julia set of the family λ exp(z)/z with real parameters λ. We look for what values of real parameters λ such that the Julia set of λ exp(z)/z does not coincide with the whole plane, and thus gives a complete classification for real parameters, which is similar to Jang’s result of a family of transcendental entire functions. Moreover, We also discuss the shape and size of Fatou sets and Julia sets of λ exp(z)/z with real parameters λ when the Julia sets are not the whole plane.

In this paper, we consider the second order Hamiltonian system -u "(t) - A(t)u(t) = del H(t, u(t)) + del W(t, u(t)), t is an element of R, where H(t, u) and W(t, u) are all even and have subquadratic growth at infinity in u. Here, we assume that 0 belongs to a spectral gap of the operator B := -d(2)/dt(2) - A(t) and obtain the existence of infinitely many small negative energy T-periodic solutions of this equation by a variant fountain theorem.

Let for each . It was proved that the set , which consists of all points whose -limit set with respect to the map F is not the set , has zero Lebesgue measure. In this article, we apply some inequalities in conformal geometry to estimate area and diameter of planar sets and show that the set has Hausdorff dimension two. This implies the Hausdorff dimension of differs from its topological dimension, thus is a fractal set.

AbstractIt is known that the Fatou set of the map exp(z)/zdefined on the punctured plane ℂ*is empty. We consider theM-set of λ exp(z)/zconsisting of all parameters λ for which the Fatou set of λexp(z)/zis empty. We prove that theM-set of λexp(z)/zhas infinite area. In particular, the Hausdorff dimension of theM-set is 2. We also discuss the area of complement of theM-set.

It is known that the Fatou set of the map exp(z)/z defined on the punctured plane C* is empty. We consider the M-set of lambda exp(z)/z consisting of all parameters A for which the Fatou set of lambda exp(z)/z is empty. We prove that the M-set of lambda exp(z)/z has infinite area. In particular, the Hausdorff dimension of the M-set is 2. We also discuss the area of complement of the M-set.