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In the integer and fractional quantum Hall effects, the electric
current flows through a thin layer under the strong magnetic field. The
diagonal resistance becomes very small at integer and specific fractional
filling factors where the electron scatterings are very few. Accordingly the
coherent length is large and therefore a tunneling effect of electrons may be
observed. We consider a new type of a quantum Hall device which has a narrow
potential barrier in the thin layer. Then the electrons flow with tunneling
effect through the potential barrier. When the oscillating magnetic field is applied
in addition to the constant field, the voltage steps may appear in the curve of
voltage V versus electric current I. If the voltage steps are
found in the experiment, it is confirmed that the 2D electron system yields the
same phenomenon as that of the ac-Josephson effect in a superconducting system.
Furthermore the step V is related to the transfer charge Q as V = (hf)/Q where f is the frequency of the
oscillating field and h is the Planck
constant. Then the detection of the step V determines the transfer charge Q. The
ratio Q/e (e is the
elementary charge) clarifies the
In this article, we study the string equation of type (2,5), which is
derived from 2D gravity theory or the string theory. We consider the equation
as a 4th order analogue of the first Painlevé equation, take the autonomous limit, and solve it
concretely by use of the Weierstrass’ elliptic function.
This article concerns the quantum
superintegrable system obtained by Tremblay and Winternitz, which allows the separation
of variables in polar coordinates and possesses three conserved quantities with
the potential described by the sixth Painlevé equation. The degeneration procedure
from the sixth Painlvé equation to the fifth one yields another new
superintegrable system; however, the Hermitian nature is broken.
The best constant of discrete Sobolev
inequality on the truncated tetrahedron with a weight which describes 2 kinds
of spring constants or bond distances. Main results coincides with the ones of
known results by Kametaka et al. under the assumption of uniformity of the
spring constants. Since the buckyball fullerene C60 has 2 kinds of edges,
destruction of uniformity makes us proceed the application to the chemistry of fullerenes.
In this article, we study the
string equation of type (2, 2n + 1), which is derived from 2D gravity theory or
the string theory. We consider the equation as a 2n-th order analogue of the
first Painlevéequation, take the autonomous limit, and solve it concretely by
use of the Weierstrass’ elliptic function.