We find generators for the full rational loop group of ${\rm GL}(n,\mathsf{C})$ as well as for the subgroup consisting of loops that satisfy the reality condition with respect to the noncompact real form ${\rm GL}(n,\mathsf{R})$. We calculate the dressing action of some of those generators on the positive loop group, and apply this to the ZS-AKNS flows and the $n$-dimensional system associated to ${\rm GL}(n,\mathsf{R})/{\rm O}(n)$.
We study the Iwasawa-type decomposition of an open subset of SL(n,ℂ)as SU(p,q)AN. We show that the dressing action of SU(p,q)is globally defined on the space of admissible elements inAN. We also show that the space of admissible elements is a multiplicative subset ofAN. We establish a geometric criterion: the symmetrization of an admissible element maps the positive cone inℂninto itself.
The fundamental characteristics of dressing action on the polyurethane pad are
investigated via dressing by single diamond of different orientations, dressing parameters and
dressing path in this study. Experimental results show that a groove with pile-up on both side walls
forms as the diamond moves over the pad with a specific dressing depth. The resulting asperities on
the pad are strongly affected by the diamond orientation. Plowing is found to be the major
mechanism responsible for this surface topology if dressing is conducted by the face of a diamond.
On the contrary, cutting action dominates when the point of a diamond is responsible for dressing. It
is also found that dressing velocity has an insignificant effect on the groove and ridges created on
the pad. The depth of the groove is smaller than the dressing depth due to the spring back of the pad.
When the groove created is repeatedly dressed over the same track, the ridge height and groove
depth increases for each additional dressing. When two grooves cross each other, the ridges at the
four corners of the intersection grow while the depth of the overlapped area decreases. These ridges
will become the pressure enhancer of the abrasives to polish the wafer.
Electro-discharge grinding (hereafter called ED-grinding) was carried out with a trial manufactured metal bond diamond wheel containing electrically conductive diamond grits (hereafter called EC-D-grits-wheel). In this research two effects i.e. removal action of workpiece by electrical discharge machining, and an in-process dressing action of the cutting edges on the grits are expected to take place. The results of ED-grinding with EC-D-grits-wheel (f100mm, SDE120Q80M) on tungsten carbide indicated a significant decrease of 21% in the grinding force, when the set discharge current was increased from 0A to 12A. It was also clarified from the alternative-grinding test with and without an electro-discharge action that stable grinding characteristics along with a reduced grinding force could be achieved in the case of the EC-D-grits-wheel with the electro-discharge action. High-speed camera photographs indicated that a stable discharge condition was achieved.
A new apparatus for microscopic observation and tracing of cutting edges of a wheel has been developed. The use of this apparatus allows us to visually witness the behavior of abrasive grains during the grinding operation. A phase-contrast microscope, as well as an electron microscope has been used with the new apparatus in this investigation. A method of calculating effective grain spacing to show distribution of cutting edges, as well as the cutting edge ratio to known worn conditions of grains has been completed. These aforementioned values for various grinding wheels have been studied in grinding operations and the changing aspects of them have been noted. Through these experiments, we have had helpful suggestions about grinding mechanisms; there is no self-dressing action of grains in the usual grinding work; sliding of the cutting edges takes place in the grinding, and wear of the grinding is increased by this action.
Generating Rational Loop Groups with Noncompact Reality Conditions