For a real analytic periodic function 𝜙: ℝ→ℝ, an integer 𝑏≥2 and 𝜆∈(1/𝑏,1), we prove the following dichotomy for the Weierstrass-type function 𝑊(𝑥)=∑_{𝑛≥0} 𝜆^𝑛 𝜙(𝑏^𝑛 𝑥): Either W(x) is real analytic, or the Hausdorff dimension of its graph is equal to 2+log𝑏𝜆. Furthermore, given b and 𝜙, the former alternative only happens for finitely many 𝜆 unless 𝜙 is constant.

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The GKZ system for the Hesse pencil of elliptic curves has more solutions than the period integrals. In this work we give different realizations and interpretations of the extra solution, in terms of oscillating integral, Eichler integral, chain integral on the elliptic curve, limit of a period of a certain compact Calabi-Yau threefold geometry, etc. We also highlight the role played by the orbifold singularity on the moduli space and its relation to the GKZ system.

Let $\mathbb {D}\subset \mathbb {C}$ be the unit disk and let $\gamma_{1},\gamma _{2}:[0,T]\to\overline{\mathbb {D}}\setminus\{0\}$ be parametrizations of two slits $\Gamma_{1}:=\gamma(0,T], \Gamma_{2}:=\gamma_{2}(0,T]$ such that $\Gamma_{1}$ and $\Gamma_{2}$ are disjoint.

We verify a long-standing conjecture of Nazarov, Treil, and Volberg.

We define piecewise continuous almost automorphic (p.c.a.a.) functions in the manners of Bochner, Bohr and Levitan, respectively, to describe almost automorphic motions in impulsive systems, and prove that with certain prefixed possible discontinuities they are equivalent to quasi-uniformly continuous Stepanov almost automorphic ones. Spatially almost automorphic sets on the line, which serve as suitable objects containing discontinuities of p.c.a.a. functions, are characterized in the manners of Bochner, Bohr and Levitan, respectively, and shown to be equivalent. Two Favard's theorems are established to illuminate the importance and convenience of p.c.a.a. functions in the study of almost periodically forced impulsive systems.

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No abstract uploaded!

No abstract uploaded!