Our study identifies a subtle deviation from Newton’s third law in the derivation of the ideal rocket equation, also known as the Tsiolkovsky Rocket Equation (TRE). TRE can be derived using a 1D elastic collision mod...Our study identifies a subtle deviation from Newton’s third law in the derivation of the ideal rocket equation, also known as the Tsiolkovsky Rocket Equation (TRE). TRE can be derived using a 1D elastic collision model of the momentum exchange between the differential propellant mass element (dm) and the rocket final mass (m1), in which dm initially travels forward to collide with m1 and rebounds to exit through the exhaust nozzle with a velocity that is known as the effective exhaust velocity ve. We observe that such a model does not explain how dm was able to acquire its initial forward velocity without the support of a reactive mass traveling in the opposite direction. We show instead that the initial kinetic energy of dm is generated from dm itself by a process of self-combustion and expansion. In our ideal rocket with a single particle dm confined inside a hollow tube with one closed end, we show that the process of self-combustion and expansion of dm will result in a pair of differential particles each with a mass dm/2, and each traveling away from one another along the tube axis, from the center of combustion. These two identical particles represent the active and reactive sub-components of dm, co-generated in compliance with Newton’s third law of equal action and reaction. Building on this model, we derive a linear momentum ODE of the system, the solution of which yields what we call the Revised Tsiolkovsky Rocket Equation (RTRE). We show that RTRE has a mathematical form that is similar to TRE, with the exception of the effective exhaust velocity (ve) term. The ve term in TRE is replaced in RTRE by the average of two distinct exhaust velocities that we refer to as fast-jet, vx<sub>1</sub>, and slow-jet, vx<sub>2</sub>. These two velocities correspond, respectively, to the velocities of the detonation pressure wave that is vectored directly towards the exhaust nozzle, and the retonation wave that is initially vectored in the direction of rocket propagation, but subsequently becomes reflected from the thrust surface of the combustion chamber to exit through the exhaust nozzle with a time lag behind the detonation wave. The detonation-retonation phenomenon is supported by experimental evidence in the published literature. Finally, we use a convolution model to simulate the composite exhaust pressure wave, highlighting the frequency spectrum of the pressure perturbations that are generated by the mutual interference between the fast-jet and slow-jet components. Our analysis offers insights into the origin of combustion oscillations in rocket engines, with possible extensions beyond rocket engineering into other fields of combustion engineering.展开更多
This article introduces a new class of ideals, namely, the sπr ideals. It is shown that every regular square matrix over sπr ideals of a ring admits a diagonal reduction.
Ideal point method is one of the methods to solve multi-objective problem. It is applied to forest harvest regu-lation, and showed very good results by analyzing changes of quantitative indexes of forest resource stru...Ideal point method is one of the methods to solve multi-objective problem. It is applied to forest harvest regu-lation, and showed very good results by analyzing changes of quantitative indexes of forest resource structure before andafter the regulation. This method can be applied as one of the mathematical tools in forest harvest regulation.展开更多
It is shown that in the Earth’s atmosphere, due to influence of the gravitational field, coefficient of thermal expansion depends on altitude. The altitude intervals for individual gases for which the laws of ideal g...It is shown that in the Earth’s atmosphere, due to influence of the gravitational field, coefficient of thermal expansion depends on altitude. The altitude intervals for individual gases for which the laws of ideal gases can be applied have been determined and it has been established that they are dependent on the adiabatic index and molar mass of these gases.展开更多
In this paper, we consider the Goldbach＇s problem for matrix rings, namely, we decompose an n × n （n 〉 1） matrix over a principal ideal domain R into a sum of two matrices in Mn,（R） with given determinants....In this paper, we consider the Goldbach＇s problem for matrix rings, namely, we decompose an n × n （n 〉 1） matrix over a principal ideal domain R into a sum of two matrices in Mn,（R） with given determinants. We prove the following result： Let n 〉 1 be a natural number and A = （aij） be a matrix in Mn（R）. Define d（A） ：= g.c.d{aij}. Suppose that p and q are two elements in R. Then （1） If n 〉 1 is even, then A can be written as a sum of two matrices X, Y in Mn（R） with det（X） = p and det（Y） = q if and only if d（A） [ p - q; （2） If n 〉 1 is odd, then A can be written as a sum of two matrices X, Y in Mn（R） with det（X） = p and det（Y） = q if and only if d（A） ｜ p ＋ q. We apply the result to the matrices in Mn（Z） and Mn（Q[x]） and prove that if R = 7. or Q[x], then any nonzero matrix A in Mn（R） can be written as a sum of two matrices in Mn（R） with prime determinants.展开更多
文摘Our study identifies a subtle deviation from Newton’s third law in the derivation of the ideal rocket equation, also known as the Tsiolkovsky Rocket Equation (TRE). TRE can be derived using a 1D elastic collision model of the momentum exchange between the differential propellant mass element (dm) and the rocket final mass (m1), in which dm initially travels forward to collide with m1 and rebounds to exit through the exhaust nozzle with a velocity that is known as the effective exhaust velocity ve. We observe that such a model does not explain how dm was able to acquire its initial forward velocity without the support of a reactive mass traveling in the opposite direction. We show instead that the initial kinetic energy of dm is generated from dm itself by a process of self-combustion and expansion. In our ideal rocket with a single particle dm confined inside a hollow tube with one closed end, we show that the process of self-combustion and expansion of dm will result in a pair of differential particles each with a mass dm/2, and each traveling away from one another along the tube axis, from the center of combustion. These two identical particles represent the active and reactive sub-components of dm, co-generated in compliance with Newton’s third law of equal action and reaction. Building on this model, we derive a linear momentum ODE of the system, the solution of which yields what we call the Revised Tsiolkovsky Rocket Equation (RTRE). We show that RTRE has a mathematical form that is similar to TRE, with the exception of the effective exhaust velocity (ve) term. The ve term in TRE is replaced in RTRE by the average of two distinct exhaust velocities that we refer to as fast-jet, vx<sub>1</sub>, and slow-jet, vx<sub>2</sub>. These two velocities correspond, respectively, to the velocities of the detonation pressure wave that is vectored directly towards the exhaust nozzle, and the retonation wave that is initially vectored in the direction of rocket propagation, but subsequently becomes reflected from the thrust surface of the combustion chamber to exit through the exhaust nozzle with a time lag behind the detonation wave. The detonation-retonation phenomenon is supported by experimental evidence in the published literature. Finally, we use a convolution model to simulate the composite exhaust pressure wave, highlighting the frequency spectrum of the pressure perturbations that are generated by the mutual interference between the fast-jet and slow-jet components. Our analysis offers insights into the origin of combustion oscillations in rocket engines, with possible extensions beyond rocket engineering into other fields of combustion engineering.
文摘This article introduces a new class of ideals, namely, the sπr ideals. It is shown that every regular square matrix over sπr ideals of a ring admits a diagonal reduction.
文摘Ideal point method is one of the methods to solve multi-objective problem. It is applied to forest harvest regu-lation, and showed very good results by analyzing changes of quantitative indexes of forest resource structure before andafter the regulation. This method can be applied as one of the mathematical tools in forest harvest regulation.
文摘It is shown that in the Earth’s atmosphere, due to influence of the gravitational field, coefficient of thermal expansion depends on altitude. The altitude intervals for individual gases for which the laws of ideal gases can be applied have been determined and it has been established that they are dependent on the adiabatic index and molar mass of these gases.
文摘In this paper, we consider the Goldbach＇s problem for matrix rings, namely, we decompose an n × n （n 〉 1） matrix over a principal ideal domain R into a sum of two matrices in Mn,（R） with given determinants. We prove the following result： Let n 〉 1 be a natural number and A = （aij） be a matrix in Mn（R）. Define d（A） ：= g.c.d{aij}. Suppose that p and q are two elements in R. Then （1） If n 〉 1 is even, then A can be written as a sum of two matrices X, Y in Mn（R） with det（X） = p and det（Y） = q if and only if d（A） [ p - q; （2） If n 〉 1 is odd, then A can be written as a sum of two matrices X, Y in Mn（R） with det（X） = p and det（Y） = q if and only if d（A） ｜ p ＋ q. We apply the result to the matrices in Mn（Z） and Mn（Q[x]） and prove that if R = 7. or Q[x], then any nonzero matrix A in Mn（R） can be written as a sum of two matrices in Mn（R） with prime determinants.
