Balanced numerical semigroups

We investigate the Frobenius number, genus, type, and minimal presentation of a class of numerical semigroups of embedding dimension 4 of the form \(S = \langle a_1, a_2, a_3, a_4 \rangle \) such that \(a_1 + a_4 = a_2 + a_3\). The investigation focuses on determining the Apery set of S with respect to the multiplicity.     

Enumerating transformation semigroups

We describe general methods for enumerating subsemigroups of finite semigroups and techniques to improve the algorithmic efficiency of the calculations. As a particular application we use our algorithms to enumerate all transformation semigroups up to degree 4. Classification of these semigroups up to conjugacy, isomorphism and anti-isomorphism, by size and rank, provides a solid base for further investigations of transformation semigroups.     

Definability of Frobenius orbits and a result on rational distance sets

We prove that the first order theory of (possibly transcendental) meromorphic functions of positive characteristic \(p>2\) is undecidable. We also establish a negative solution to an analogue of Hilbert’s tenth problem for such fields of meromorphic functions, for Diophantine equations including vanishing conditions. These undecidability results are proved by showing that the binary relation \(\exists s\ge 0, f=g^{p^s}\) is positive existentially definable in such fields. We also prove that the abc conjecture implies a solution to the Erdös–Ulam problem on rational distance sets. These two seemingly distant topics are addressed by a study of power values of bivariate polynomials of the form F(X)G(Y).     

Continuity and analyticity for a cross-coupled Camassa–Holm equation with waltzing peakons and compacton pairs

In this paper, we mainly study the well-posedness in the sense of Hadamard, non-uniform dependence, Hölder continuity and analyticity of the data-to-solution map for a cross-coupled Camassa–Holm equation with waltzing peakons and compacton pairs on both the periodic and the nonperiodic case. Using a Galerkin-type approximation scheme, it is shown that this equation is well-posed in Sobolev spaces \(H^{s} \times H^{s},s>5/2\) in the sense of Hadamard, that is, the data-to-solution mapis continuous. In conjunction with the well-posedness estimate, it is also proved that this dependence is sharp by showing that the solution map is not uniformly continuous. Furthermore, the Hölder continuous in the \(H^r \times H^r\) topology when \(0\le r< s\) with Hölder exponent \(\alpha \) depending on both s and r are shown. Finally, applying generalized Ovsyannikov type theorem and the basic properties of Sobolev-Gevrey spaces, we prove the Gevrey regularity and analyticity of the CCCH system. Moreover, we obtain a lower bound of the lifespan and the continuity of the data-to-solution map     

Multifractal analysis of the divergence points of Birkhoff averages for $$\beta $$-transformations

This paper is aimed at a detailed study of the multifractal analysis of the so-called divergence points in the system of \(\beta \)-expansions. More precisely, let \(T_{\beta }\) be the \(\beta \)-transformation on [0, 1) for a general \(\beta >1\) and \(\psi :[0,1]\mapsto \mathbb {R}\) be a continuous function. Denote by \(\textsf {A}(\psi ,x)\) all the accumulation points of \(\{\frac{1}{n}\sum _{j=0}^{n-1}\psi (T^jx): n\ge 1\}\). The Hausdorff dimensions of the sets

$$\begin{aligned} \{x:\textsf {A}(\psi ,x)\supset [a,b]\},\quad \{x:\textsf {A}(\psi ,x)=[a,b]\},\quad \{x:\textsf {A}(\psi ,x)\subset [a,b]\} \end{aligned}$$

i.e., the points for which the Birkhoff averages of \(\psi \) do not exist but behave in a certain prescribed way, are determined completely for any continuous function \(\psi \).

     

Some evaluation of q-analogues of Euler sums

In this paper, we discuss the analytic representations of q-Euler sums which involve q-harmonic numbers through q-polylogarithms, either linearly or nonlinearly, and give explicit formulae for several classes of q-Euler sums in terms of q-polylogarithms and q-special functions. Furthermore, we develop new closed form representations of sums of quadratic and cubic parametric q-Euler sums. Finally, we can find that the q-Euler sums are reducible to the classical Euler sums when q approaches 1.     

Oscillation of holomorphic Bergman–Besov kernels on the ball

We estimate the oscillation of holomorphic Bergman–Besov reproducing kernels on the unit ball of \(\mathbb {C}^n\). As an application of this estimate we characterize holomorphic Bergman–Besov spaces \(A_\alpha ^p\,(\alpha \in \mathbb {R})\) in terms of double integrals of the fractions \(|f(z)-f(w)|/|z-w|\) and \(|f(z)-f(w)|/|1-\langle z,w \rangle |\) and complete the earlier works done on this subject. Our results provide, when \(\alpha \le -1\), a derivative-free characterization of \(A_\alpha ^p\).     

The dimensions of the divergence points of self-similar measures with weak separation condition

Let \(\mu \) be the self-similar measure supported on the self-similar set K with the weak separation condition, which is weaker than the open set condition. This article uses Hausdorff dimension and packing dimension to investigate the multifractal structure of several sets of divergence points of \(\mu \) in the iterated function system.     

Spectrality of infinite convolutions with three-element digit sets

Let \(0< \rho <1\) and let \(\{a_n, b_n\}_{n=1}^\infty \) be a sequence of integers with bounded from upper and lower. Associated with them there exists a unique Borel probability measure \(\mu _{\rho , \{0, a_n, b_n\}}\) generated by the following infinite convolution product

$$\begin{aligned} \mu _{\rho , \{0, a_n, b_n\}}=\delta _{\rho \{0, a_1, b_1\}} *\delta _{\rho ^2 \{0, a_2, b_2\}} *\delta _{\rho ^3 \{0, a_3, b_3\}} *\cdots \end{aligned}$$

in the weak convergence, where \(\delta _E=\frac{1}{\# E}\sum _{e \in E} \delta _e\) and \(\hbox {gcd}(a_n, b_n)=1\) for all \(n \in {{\mathbb {N}}}\). In this paper, we show that \(L^2(\mu _{\rho , \{0, a_n, b_n\}})\) admits an exponential orthonormal basis if and only if \(\rho ^{-1} \in 3{{\mathbb {N}}}\) and  \(\{a_n, b_n\} \equiv \{1, 2\} \ (\mathrm {mod} \ 3)\) for all \(n \in {{\mathbb {N}}}\).

     

Equality statements for Schwarz–Pick type estimates and the pseudo-hyperbolic distance

In the present article we demonstrate that the conclusion on the case of equality for the estimate of Schwarz–Pick type is a property of a distance-nonincreasing holomorphic mapping which is locally defined. Also, while considering existence of branch points, we deal with the topics in terms of the pseudo-hyperbolic distance.