Throughout our study of the enumeration of plane partitions we make use of bijective proofs to find generating functions. In particular, we consider bounded plane partitions, symmetric plane partitions and weak reverse plane partitions. Using the combinatorial interpretations of Schur functions in relation to semistandard Young tableaux, we rely on the properties of symmetric functions. In our ...

Almost all known restrictions on the topology of nonsingular real algebraic curves in the projective plane are also valid for a wider class of objects: real pseudo-holomorphic curves. It is still unknown if there exists a nonsingular real pseudo-holomorphic curve not isotopic in the projective plane to a real algebraic curve of the same degree. In this article, we focus our study on symmetric r...

Orientation of cell division is a vital aspect of tissue morphogenesis and growth. Asymmetric divisions generate cell fate diversity and epithelial stratification, whereas symmetric divisions contribute to tissue growth, spreading, and elongation. Here, we describe a mechanism for positioning the spindle in symmetric cell divisions of an embryonic epithelium. We show that during the early stage...

An Au-SiO2-Au (metal-insulator-metal; MIM) surface-plasmon-polariton (SPP) guide is schematically presented in Fig. S1a. As the name “MIM” implies, a structural symmetry exists with respect to the mirror plane that cuts through the center of the SiO2 layer and is parallel to the x–y plane. As a result, supported SPP modes in the guide can be classified into two major types, each with its own di...

it is well known fact that binary relations are generalized mathematical functions. contrary to functions from domain to range, binary relations may assign to each element of domain two or more elements of range. some basic operations on functions such as the inverse and composition are applicable to binary relations as well. depending on the domain or range or both are fuzzy value fuzzy set, i...

the non-commuting graph $nabla(g)$ of a non-abelian group $g$ is defined as follows: its vertex set is $g-z(g)$ and two distinct vertices $x$ and $y$ are joined by an edge if and only if the commutator of $x$ and $y$ is not the identity. in this paper we 'll prove that if $g$ is a finite group with $nabla(g)congnabla(bs_{n})$, then $g cong bs_{n}$, where $bs_{n}$ is the symmetric group of degre...

By generalizing the wave aberration function to include plane symmetric systems, we describe the aberration fields for a combination of plane symmetric systems. The combined system aberration coefficients for the fields of spherical aberration, coma, astigmatism, defocus and distortion depend on the individual aberration coefficients and the orientations of the individual plane symmetric compon...

background: limited resources for adult stem cells necessitate their in vitro culture prior to clinical use. investigating mitochondrial dna (mtdna) and telomere shortening has proved to be important indications of stem cell validity. this study was designed to investigate these indicators in multiple passages of three adult stem cell lines which were produced in our stem cell laboratory. metho...