In this work, we will propose some improvements and innovations for Submerged Combustion, namely a Pressurized Submerged Combustion Technology. This paper introduces the Pressurized Submerged Combustion and some basic theoretical research.To understand better gas pressure fluctuations and their influence in flame stability, a mathematical model is developed in this paper, and its particular solution is elucidated. The results show that, the variations of pressure are influenced by bubbling which is a self-excited oscillation phenomenon. Increasing the damping of liquid motion is very helpful for stabilizing pressure fluctuations. The model can be used for guidance and optimization design of Pressurized Submerged Combustion equipments.

In this article, we discuss the growth of entire functions represented by Laplace–Stieltjes transform converges on the whole complex plane and obtain some equivalence conditions about proximate growth of Laplace–Stieltjes transforms with finite order and infinite order. In addition, we also investigate the approximation of Laplace–Stieltjes transform with the proximate order and obtain some results containing the proximate growth order, the error, An∗, and λn, which are the extension and improvement of the previous theorems given by Luo and Kong and Singhal and Srivastava.

By utilizing the Nevanlinna theory of meromorphic functions in several complex variables, we mainly investigate the existence and the forms of entire solutions for the partial differential-difference equation of Fermat type α ∂ f ( z 1 , z 2 ) ∂ z 1 + β ∂ f ( z 1 , z 2 ) ∂ z 2 m + f ( z 1 + c 1 , z 2 + c 2 ) n = 1 {\left(\alpha \frac{\partial f\left({z}_{1},{z}_{2})}{\partial {z}_{1}}+\beta \frac{\partial f\left({z}_{1},{z}_{2})}{\partial {z}_{2}}\right)}^{m}+f{\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})}^{n}=1 and α ∂ f ( z 1 , z 2 ) ∂ z 1 + β ∂ f ( z 1 , z 2 ) ∂ z 2 2 + [ γ 1 f ( z 1 + c 1 , z 2 + c 2 ) − γ 2 f ( z 1 , z 2 ) ] 2 = 1 , {\left(\alpha \frac{\partial f\left({z}_{1},{z}_{2})}{\partial {z}_{1}}+\beta \frac{\partial f\left({z}_{1},{z}_{2})}{\partial {z}_{2}}\right)}^{2}+{\left[{\gamma }_{1}f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})-{\gamma }_{2}f\left({z}_{1},{z}_{2})]}^{2}=1, where m , n m,n are positive integers and α , β , γ 1 , γ 2 \alpha ,\beta ,{\gamma }_{1},{\gamma }_{2} are constants in C {\mathbb{C}} . We give some results about the forms of solutions for these equations, which are great improvements of the previous theorems given by Xu and Cao et al. Moreover, it is very satisfactory that we give the corresponding examples to explain the conclusions of our theorems in each case.