On zeros and growth of solutions of complex difference equations

Let f be an entire function of finite order, let $n\geq 1$ n ≥ 1 , $m\geq 1$ m ≥ 1 , $L(z,f)\not \equiv 0$ L ( z , f ) ≢ 0 be a linear difference polynomial of f with small meromorphic coefficients, and $P_{d}(z,f)\not \equiv 0$ P d ( z , f ) ≢ 0 be a difference polynomial in f of degree $d\leq n-1$ d ≤ n − 1 with small meromorphic coefficients. We consider the growth and zeros of $f^{n}(z)L^{m}(z,f)+P_{d}(z,f)$ f n ( z ) L m ( z , f ) + P d ( z , f ) . And some counterexamples are given to show that Theorem 3.1 proved by I. Laine (J. Math. Anal. Appl. 469:808–826, 2019) is not valid. In addition, we study meromorphic solutions to the difference equation of type $f^{n}(z)+P_{d}(z,f)=p_{1}e^{\alpha _{1}z}+p_{2}e^{\alpha _{2}z}$ f n ( z ) + P d ( z , f ) = p 1 e α 1 z + p 2 e α 2 z , where $n\geq 2$ n ≥ 2 , $P_{d}(z,f)\not \equiv 0$ P d ( z , f ) ≢ 0 is a difference polynomial in f of degree $d\leq n-2$ d ≤ n − 2 with small mromorphic coefficients, $p_{i}$ p i , $\alpha _{i}$ α i ($i=1,2$ i = 1 , 2 ) are nonzero constants such that $\alpha _{1}\neq \alpha _{2}$ α 1 ≠ α 2 . Our results are improvements and complements of Laine 2019, Latreuch 2017, Liu and Mao 2018.

Velocity and Diameter Measurements of Penetrating Arteries by Model Based Analysis of Complex Difference Images in Phase Contrast MRI

Pathological changes of penetrating arteries (PAs) within deep white matter (WM) may be an important contributing factor of cerebral small vessel disease (SVD). Quantitative characterization of the PAs is important for further illuminating their roles in SVD but remains challenging due to their sub-voxel sizes. We propose a quantitative MRI approach for measuring the diameters and flow velocities of PAs based on model based analysis of complex difference images in phase contrast MRI. The complex difference image of each PA is fitted by a model image calculated by taking into account the partial volume effect and signal enhancement due to in flow effects to obtain velocity , diameter (D), and volume flow rate (VFR) of the PAs. Simulation, phantom, and in vivo studies were carried out to evaluate the accuracy and measurement errors of the proposed method. Our results suggest that PAs with velocities ≥ 0.8 cm/s can be accurately measured with , D, and VFR errors of 0.28 cm/s, 20 μm, and 0.024 mm3/s, respectively, although the mean lumen area occupies only 18% of the acquired pixel area. The PAs have a distribution peak at ~1.2 cm/s and diameters distribution mostly in the range of 88 – 200 μm Quantitative measurements of PAs with the MBAC method may serve as an invaluable tool for illuminating the role of PAs in the aetiopathogenesis of cerebral SVD.