Shock waves in Euler equations for compressible medium

Shock waves of arbitrary strength in the Euler equations for compressible media are studied. The admissibility condition for a shock wave is shown to be equivalent to its formation according to the entropy production criterion. The Riemann problem with large data has a unique admissible solutions. These quantitative results are based on the exact global expressions for the basic physical variables as the states move along the Hugoniot and wave curves.

An Almost Comprehensive Approach for the Choice of Motor and Transmission in Mechatronic Applications: Torque Peak of the Motor

The choice of motor and transmission to move a joint must ensure that the torque peaks of the motor lie inside its dynamic operating range. With this aim, this paper proposes an approach in which all the candidate transmissions are processed one by one to find among all the candidate motors those they could execute the reference task with. Consequently, all the transmission parameters, and not only its transmission ratio, are taken into consideration in advance. For rectangular dynamic operating ranges, this approach allows a direct and precise evaluation of all the admissible motor-transmission couples, without any approximation and further check. Apart from an entirely automated procedure, the method also provides diagrams through which the designer can concisely compare the admissible solutions. Furthermore, the method provides a solution for the drive systems in which the limit torque of the dynamic operating range does depend on the motor speed.

Global behavior of a third order difference equation with quadratic term

In this paper, we solve and study the global behavior of the admissible solutions of the difference equation $$\begin{aligned} x_{n+1}=\frac{x_{n}x_{n-2}}{-ax_{n-1}+bx_{n-2}}, \quad n=0,1,\ldots , \end{aligned}$$ x n + 1 = x n x n - 2 - a x n - 1 + b x n - 2 , n = 0 , 1 , … , where $$a, b>0$$ a , b > 0 and the initial values $$x_{-2}$$ x - 2 , $$x_{-1}$$ x - 1 , $$x_{0}$$ x 0 are real numbers.

The Neumann problem of Hessian quotient equations

In this paper, we obtain some important inequalities of Hessian quotient operators, and global [Formula: see text] estimates of the Neumann problem of Hessian quotient equations. By the method of continuity, we establish the existence theorem of [Formula: see text]-admissible solutions of the Neumann problem of Hessian quotient equations.

On the number of linearly independent admissible solutions to linear differential and linear difference equations

A classical theorem of Frei states that if $A_p$ is the last transcendental function in the sequence $A_0,\ldots ,A_{n-1}$ of entire functions, then each solution base of the differential equation $f^{(n)}+A_{n-1}f^{(n-1)}+\cdots +A_{1}f'+A_{0}f=0$ contains at least $n-p$ entire functions of infinite order. Here, the transcendental coefficient $A_p$ dominates the growth of the polynomial coefficients $A_{p+1},\ldots ,A_{n-1}$ . By expressing the dominance of $A_p$ in different ways and allowing the coefficients $A_{p+1},\ldots ,A_{n-1}$ to be transcendental, we show that the conclusion of Frei’s theorem still holds along with an additional estimation on the asymptotic lower bound for the growth of solutions. At times, these new refined results give a larger number of linearly independent solutions of infinite order than the original theorem of Frei. For such solutions, we show that $0$ is the only possible finite deficient value. Previously, this property has been known to hold for so-called admissible solutions and is commonly cited as Wittich’s theorem. Analogous results are discussed for linear differential equations in the unit disc, as well as for complex difference and complex q-difference equations.

Three-point bending test of pantographic blocks: numerical and experimental investigation

The equilibrium forms of pantographic blocks in a three-point bending test are investigated via both experiments and numerical simulations. In the computational part, the corresponding minimization problem is solved with a deformation energy derived by homogenization within a class of admissible solutions. To evaluate the numerical simulations, series of measurements have been carried out with a suitable experimental setup guided by the acquired theoretical knowledge. The observed experimental issues have been resolved to give a robust comparison between the numerical and experimental results. Promising agreement between theoretical predictions and experimental results is demonstrated for the planar deformation of pantographic blocks.