Abstract: In this paper we show that there are applications that transform the movement of a pendulum into movements in R3. This can be done using Euler top system of differential equations. On the constant level surfaces, Euler top system reduces to the equation of a pendulum. Those properties are also considered in the case of system of differential equations with delay argument and in the fr...

We have obtained nonperturbative one-loop expressions for the mean energy-momentum tensor and current density of Dirac's field on a constant electric-like background. One of the goals of this calculation is to give a consistent description of back-reaction in such a theory. Two cases of initial states are considered: the vacuum state and the thermal equilibrium state. First, we perform calculat...

We prove a formula relating the analytic torsion and Reidemeister torsion on manifolds with boundary in the general case when the metric is not necessarily a product near the boundary. The product case has been established by W. Luc̈k and S. M. Vishik. We find that the extra term that comes in here in the nonproduct case is the transgression of the Euler class in the even dimensional case and a ...

ure of the nonmonotonicity of f. In particular, F is identically zero if and only if f is strictly increasing . Here we shall take f to be (p, Euler's function, and study the associated function F 4„ which we henceforth call F. We shall show that F(n)/n is asymptotically representable as a function of T(n)/n . Then we shall prove that F(n)/n has a distribution function. We shall study max,,, F(...

Let φ(n) be the Euler function of n. We prove that there are at most finitely many composite integers n such that φ(n) | n − 1 and P (φ(n)) ≡ 0 (mod n), where P (X) ∈ Z[X] is any monic non-constant polynomial.

In this paper, we are concerned with the Cauchy problem on the compressible isentropic two-fluids Euler-Maxwell equations in three dimensions. The global existence of solutions near constant steady states with the vanishing electromagnetic field is established, and also the time-decay rates of perturbed solutions in L space for 2 ≤ q ≤ ∞ are obtained. The proof for existence is due to the class...

Any rotation of a 3-dimensional object can be performed by three consecutive rotations over Euler angles. Intrinsic rotations produce the same result as extrinsic rotations in transformation. Euler rotations are the movement obtained by changing one of the Euler angles while leaving the other two constant. These Euler rotations are applied in a simple two-axis gimbals set mounted on an automoti...

In this paper we introduce the (I, J) similar method for incompressible two-dimensional Euler equations, and obtain a series of explicit (I, J) similar solutions to the incompressible two-dimensional Euler equations. These solutions include all of the twin wave solutions, some new singularity solutions, and some global smooth solutions with a finite energy. We also reveal that the twin wave sol...