In this article we propose the integral, series and product representations for the Ramanujan zeta-function. We suggest a variant for the Conrey-Ghosh product for the entire Ramanujan zeta-function. We present some variants for the product for the Ramanujan $\Xi$-function. We prove that all of its zeros are real. Along the way we obtain the truth of the Riemann-Hardy conjecture.

In this paper we address some variants for the products of Hadamard and Patterson. We prove that all zeros of the Riemann $\Xi$--function are real. We also prove that the Riemann hypothesis is true. The equivalence theorems associated with the Riemann zeta--function are obtained in detail.

The present paper deals with a class of even entire functions of order $\rho =1$ and genus $\vartheta =0$ of the polynomials form, \[ \sum\limits_{m=0}^\infty {\frac{\left( {-1} \right)^m\Phi ^{\left( {2m} \right)}\left( 0 \right)}{\Gamma \left( {2m+1} \right)}x^{2m}} =\Phi \left( 0 \right)\prod\limits_{k=1}^\infty {\left( {1-\frac{x}{\ell _k }} \right)} , \] where $\Phi\left( 0 \right)\ne 0$, real numbers $x$, nonnegative integers $m$, and $\ell _k \ne 0$ are all of the nonzero roots with $\sum\limits_{k=1}^\infty {1/\left| {\ell _k } \right|} <\infty $ and natural numbers $k$. We provide an efficient criterion for the polynomials with only real zeros. We also prove that the conjecture of Jensen is our special case.

YH-DIE must have continuity . Given the basic algebraic clusters of homogeneous configurations,we can get the basic three equations: \begin{array}{l} {\mathop{\int}\nolimits_{0}\nolimits^{{x}_{i}}{\frac{{G}\left({{x}_{i}\mathrm{,}s}\right)}{{\left({{x}_{i}\mathrm{{-}}{s}}\right)}^{\mathit{\alpha}}}\mathrm{\varphi}\left({s}\right){ds}}\mathrm{{=}}{f}\left({{x}_{i}}\right)}\ ;\ {\frac{\mathrm{\partial}}{\mathrm{\partial}{x}_{i}}\left({\frac{{\mathrm{\partial}}_{{x}_{i}}G}{\sqrt{{1}\mathrm{{+}}{\left|{\mathrm{\nabla}{G}}\right|}^{2}}}}\right)\mathrm{{=}}{0}}\ ;\ {{i}\mathrm{{=}}\mathop{\sum}\limits_{{x}_{i}\mathrm{{=}}{1}}\limits^{\mathrm{\infty}}{\arccos\hspace{0.33em}\mathrm{\varphi}\left({{x}_{i}}\right)}\mathrm{{=}}{f}\left({\fbox{${Yuh}$}}\right)} \end{array} YH-DIE has become a fusion point and access point in the fields of algebraic geometry and partial differential equations, and its mapping on multidimensional algebraic clusters or manifolds is very special. The minimal surface equation is a special case.

In this paper, we study the entire or meromorphic solutions for differential-difference equations in f(z) , its shifts, its derivatives and derivatives of its shifts. and study some Hayman's results for differential-difference polynomials.

We investigate the value distribution of dierence polynomials of entire and meromorphic functions, which can be viewed as the Hayman's conjecture. And we also study the uniqueness of dierence polynomials sharing a common value.

In this paper, we consider the zero distributions of q-shift monomi-als and difference polynomials of meromorphic functions with zero order, that extends the classical Hayman results on the zeros of differential poly-nomials to q-shift difference polynomials. We also investigate problem of q-shift difference polynomials that share a common value.

In modern warfare, the diversified tasks can sometimes be challenging to fulfill with a single projectile, necessitating the incorporation of cooperation constraints in trajectory planning. Limited by the stringent requirements of the launch platform and lower costs, the maneuverability and ability to adjust the flight time range of projectiles are weak. Therefore, the rationality of performance indicators and flight time coordination strategies is especially critical. To fully exploit the control capabilities of projectiles, a battlefield area segmentation method is proposed, differentiated performance indicators and a cooperative trajectory planning model that considers the entire range of projectiles are established. Considering the limitations of existing flight time coordination strategies in applications, targeted improvements have been made in different cooperation scenarios, and a bi-level adaptive cooperation strategy(BACS) with both optimality and flexibility is proposed. To rapidly determine feasible conditions, a simple and universal method for obtaining the capability ranges of cooperation is introduced. The multi-phase optimal control problems are transformed into nonlinear programming problems using the Radau pseudo-spectral method, and simulation results show that the proposed methods are capable of handling trajectory planning tasks in different cooperation scenarios. The improved strategies effectively enhance the planning success rate under relatively harsh conditions and BACS demonstrates good compatibility with the problems studied.