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every partition and its conjugation , there is an important invariant , which denotes the number of different parts. That is , . We will derive a series of symmetric q-identities from the invariant in partition conjugation by
studying modified Durfee rectangles. The extensive applications of the several
symmetric q-identities in q-series  will also be discussed. Without too much effort one can obtain much well-known knowledge
as well as new formulas by proper substitutions and elementary calculations,
such as symmetric identities, mock theta functions, a two-variable reciprocity
theorem, identities from Ramanujan’s Lost Notebook and so on.
films based on polyurethane and polyvinyl chloride (PU/PVC) doped by low content
(≤0.045 wt%) of single wall carbon nanotubes (SWCNTs) were prepared and characterized.
The structural, optical, thermal and mechanical properties of the prepared nanocomposites
were characterized by various tools. The IR band at 3437 cm-1 in SWCNTs
vanishing for doped samples signified that SWCNTs were successfully covalently encapsulated
with PU/PVC. The X-ray peaks broaden with reduced intensity which refers to a reduction of the crystallinity for the blend after increase of
SWCNTs attributed to changing in the crosslink density. UV/Vis analysis reveals that the optical band gap values were decreased
with increased SWCNT attributed to the formation of defects in polymeric matrix with increase
in the degree of disorder in the films. These defects produce the localized states
in the optical band gap. All results data showed well dispersion of SWCNTs in polymer
matrix at all weight percentages and an interaction between SWCNTs and the blend
occurred. The thermal analysis showed that the melting point (Tm) and degradation
temperature (Td) of the obtained films were increased gradually with
adding of SWCNTs. Furthermore, the thermal stability was improved after adding SWCNTs
into the PU/PVC polymer blend.