A Positive Answer for 3IM+1CM Problem with a General Difference Polynomial

In this paper, the 3IM+1CM theorem with a general difference polynomial L z , f will be established by using new methods and technologies. Note that the obtained result is valid when the sum of the coefficient of L z , f is equal to zero or not. Thus, the theorem with the condition that the sum of the coefficient of L z , f is equal to zero is also a good extension for recent results. However, it is new for the case that the sum of the coefficient of L z , f is not equal to zero. In fact, the main difficulty of proof is also from this case, which causes the traditional theorem invalid. On the other hand, it is more interesting that the nonconstant finite-order meromorphic function f can be exactly expressed for the case f ≡ − L z , f . Furthermore, the sharpness of our conditions and the existence of the main result are illustrated by examples. In particular, the main result is also valid for the discrete analytic functions.

Efficient and High Path Quality Autonomous Exploration and Trajectory Planning of UAV in an Unknown Environment

The ability of an autonomous Unmanned Aerial Vehicle (UAV) in an unknown environment is a prerequisite for its execution of complex tasks and is the main research direction in related fields. The autonomous navigation of UAVs in unknown environments requires solving the problem of autonomous exploration of the surrounding environment and path planning, which determines whether the drones can complete mission-based flights safely and efficiently. Existing UAV autonomous flight systems hardly perform well in terms of efficient exploration and flight trajectory quality. This paper establishes an integrated solution for autonomous exploration and path planning. In terms of autonomous exploration, frontier-based and sampling-based exploration strategies are integrated to achieve fast and effective exploration performance. In the study of path planning in complex environments, an advanced Rapidly Exploring Random Tree (RRT) algorithm combining the adaptive weights and dynamic step size is proposed, which effectively solves the problem of balancing flight time and trajectory quality. Then, this paper uses the Hermite difference polynomial to optimization the trajectory generated by the RRT algorithm. We named proposed UAV autonomous flight system as Frontier and Sampling-based Exploration and Advanced RRT Planner system (FSEPlanner). Simulation performs in both apartment and maze environment, and results show that the proposed FSEPlanner algorithm achieves greatly improved time consumption and path distances, and the smoothed path is more in line with the actual flight needs of a UAV.

Results on uniqueness of entire functions related to differential-difference polynomial

In the paper, we use the idea of normal family to investigate the uniqueness problems of entire functions when certain types of differential-difference polynomials generated by them sharing a non-zero polynomial. Also we exhibit one example to show that the conditions of our results are the best possible.

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.

Entire functions that share a small function with their linear difference polynomial

<abstract><p>In this paper, we investigate the uniqueness of an entire function sharing a small function with its linear difference polynomial. Our results improve some results due to Li and Yi ^{[<xref ref-type="bibr" rid="b11">11</xref>]}, Zhang, Chen and Huang ^{[<xref ref-type="bibr" rid="b17">17</xref>]}, Zhang, Kang and Liao ^{[<xref ref-type="bibr" rid="b18">18</xref>,<xref ref-type="bibr" rid="b19">19</xref>]} etc.</p></abstract>

Uniqueness of difference polynomials

<abstract><p>Let $ f(z) $ be a transcendental meromorphic function of finite order and $ c\in\Bbb{C} $ be a nonzero constant. For any $ n\in\Bbb{N}^{+} $, suppose that $ P(z, f) $ is a difference polynomial in $ f(z) $ such as $ P(z, f) = a_{n}f(z+nc)+a_{n-1}f(z+(n-1)c)+\cdots+a_{1}f(z+c)+a_{0}f(z) $, where $ a_{k} (k = 0, 1, 2, \cdots, n) $ are not all zero complex numbers. In this paper, the authors investigate the uniqueness problems of $ P(z, f) $.</p></abstract>

Meromorphic solutions of three certain types of non-linear difference equations

<abstract><p>In this paper, the representations of meromorphic solutions for three types of non-linear difference equations of form</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ f^{n}(z)+P_{d}(z, f) = u(z)e^{v(z)}, $\end{document} </tex-math></disp-formula></p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ f^{n}(z)+P_{d}(z, f) = p_{1}e^{\lambda z}+p_{2}e^{-\lambda z} $\end{document} </tex-math></disp-formula></p> <p>and</p> <p><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ f^{n}(z)+P_{d}(z, f) = p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z} $\end{document} </tex-math></disp-formula></p> <p>are investigated, where $ n\geq 2 $ is an integer, $ P_{d}(z, f) $ is a difference polynomial in $ f $ of degree $ d\leq n-1 $ with small coefficients, $ u(z) $ is a non-zero polynomial, $ v(z) $ is a non-constant polynomial, $ \lambda, p_{j}, \alpha_{j}\; (j = 1, 2) $ are non-zero constants. Some examples are also presented to show our results are best in certain sense.</p></abstract>