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present methods to characterize mesenchymal stromal cells (MSC) over long time
periods in vitro. The methods entail
passaging cells multiple times and performing differentiation studies with the
cells at each passage. Using an array of surface markers and flow cytometric
quantification, the data can be correlated to traditional measures of
differentiation such as PCR and staining. Using these methods to quantify the
amount of differentiation, we concluded that many common MSC markers do not
specifically define MSCs with true stem cell properties. Additionally,
adipose-derived as opposed to bone marrow-derived MSCs show long-term CD34+ labeling. The methods described can be used to help identify stem cell
markers and to characterize the state of stem cells in vitro. Compiling these data from multiple laboratories would be
helpful to determine source, extraction and culture methods needed to obtain
high yields of useful stem cells.
geometric techniques, formulas for the number of squares that require k moves in order to be reached by a
sole knight from its initial position on an infinite
chessboard are derived. The number of squares reachable in exactly k moves are 1, 8, 32, 68, and 96 for k = 0, 1, 2, 3, and 4, respectively, and
28k – 20 for k ≥ 5. The cumulative number of squares reachable in k or fever moves are 1, 9, 41, and 109 for k = 0, 1, 2, and 3, respectively, and 14k2 – 6k + 5 for k ≥ 4. Although these formulas are known, the proofs that are presented are new and more
mathematically accessible then preceding proofs.
Fitting of full X-ray diffraction patterns is an
effective method for quantifying abundances during X-ray diffraction (XRD)
analyses. The method is based on the principal that the observed diffraction
pattern is the sum of the individual phases that compose the sample. By adding
an internal standard (usually corundum) to both the observed patterns and to
those for individual pure phases (standards), all patterns can all be
normalized to an equivalent intensity based on the internal standard intensity.
Using least-squares refinement, the individual phase proportions are varied
until an optimal match is reached. As the fitting of full patterns uses the
entire pattern, including background, disordered and amorphous phases are
explicitly considered as individual phases, with their individual intensity
profiles or “amorphous humps” included in the refinement. The method can be
applied not only to samples that contain well-ordered materials, but it is
particularly well suited for samples containing amorphous and/or disordered
materials. In cases with extremely disordered materials
where no crystal structure is available for Rietveld refinement or there is no
unique intensity area that can be measured for a traditional RIR analysis,
full-pattern fitting may be the best or only way to readily obtain quantitative
results. This approach is also applicable in cases where there are several
coexisting highly disordered phases. As all phases are considered as discrete
individual components, abundances are not constrained to sum to 100%.
Generalizations of the geometric construction that repeatedly
attaches rectangles to a square, originally given by Myerson, are presented.
The initial square is replaced with a rectangle, and also the dimensionality of
the construction is increased. By selecting values for the various parameters,
such as the lengths of the sides of the original rectangle or rectangular box
in dimensions more than two and their relationships to the size of the attached
rectangles or rectangular boxes, some interesting formulas are found. Examples
are Wallis-type infinite-product formulas for the areas of p-circles with p > 1.
The normal direction to the normal direction to a line in Minkowski
geometries generally does not give the original line. We show that in lp geometries with p>1 repeatedly
finding the normal line through the origin gives sequences of lines that
monotonically approach specific lines of symmetry of the unit circle. Which
lines of symmetry that are approached depends upon the value of p and the slope of the initial line.