Abstract:
Scientific literature is increasingly reporting on dietary deficiencies in many populations of some nutrients critical for foetal and infant brain development and function. Purpose: To highlight the potential benefits of maternal supplementation with docosahexaenoic acid (DHA) and other important complimentary nutrients, including vitamin D, folic acid and iodine during pregnancy and/or breast feeding for foetal and/or infant brain development and/or function. Methods: English language systematic reviews, meta-analyses, randomised controlled trials, cohort studies, cross-sectional and case-control studies were obtained through searches on MEDLINE and the Cochrane Register of Controlled Trials from January 2000 through to February 2012 and reference lists of retrieved articles. Reports were selected if they included benefits and harms of maternal supplementation of DHA, vitamin D, folic acid or iodine supplementation during pregnancy and/or lactation. Results: Maternal DHA intake during pregnancy and/or lactation can prolong high risk pregnancies, increase birth weight, head circumference and birth length, and can enhance visual acuity, hand and eye co-ordination, attention, problem solving and information processing. Vitamin D helps maintain pregnancy and promotes normal skeletal and brain development. Folic acid is necessary for normal foetal spine, brain and skull development. Iodine is essential for thyroid hormone production necessary for normal brain and nervous system development during gestation that impacts childhood function. Conclusion: Maternal supplementation within recommended safe intakes in populations with dietary deficiencies may prevent many brain and central nervous system malfunctions and even enhance brain development and function in their offspring.

Abstract:
The $k$-Young lattice $Y^k$ is a partial order on partitions with no part larger than $k$. This weak subposet of the Young lattice originated from the study of the $k$-Schur functions(atoms) $s_\lambda^{(k)}$, symmetric functions that form a natural basis of the space spanned by homogeneous functions indexed by $k$-bounded partitions. The chains in the $k$-Young lattice are induced by a Pieri-type rule experimentally satisfied by the $k$-Schur functions. Here, using a natural bijection between $k$-bounded partitions and $k+1$-cores, we establish an algorithm for identifying chains in the $k$-Young lattice with certain tableaux on $k+1$ cores. This algorithm reveals that the $k$-Young lattice is isomorphic to the weak order on the quotient of the affine symmetric group $\tilde S_{k+1}$ by a maximal parabolic subgroup. From this, the conjectured $k$-Pieri rule implies that the $k$-Kostka matrix connecting the homogeneous basis $\{h_\la\}_{\la\in\CY^k}$ to $\{s_\la^{(k)}\}_{\la\in\CY^k}$ may now be obtained by counting appropriate classes of tableaux on $k+1$-cores. This suggests that the conjecturally positive $k$-Schur expansion coefficients for Macdonald polynomials (reducing to $q,t$-Kostka polynomials for large $k$) could be described by a $q,t$-statistic on these tableaux, or equivalently on reduced words for affine permutations.

Abstract:
We prove that structure constants related to Hecke algebras at roots of unity are special cases of k-Littlewood-Richardson coefficients associated to a product of k-Schur functions. As a consequence, both the 3-point Gromov-Witten invariants appearing in the quantum cohomology of the Grassmannian, and the fusion coefficients for the WZW conformal field theories associated to \hat{su}(\ell) are shown to be k-Littlewood Richardson coefficients. From this, Mark Shimozono conjectured that the k-Schur functions form the Schubert basis for the homology of the loop Grassmannian, whereas k-Schur coproducts correspond to the integral cohomology of the loop Grassmannian. We introduce dual k-Schur functions defined on weights of k-tableaux that, given Shimozono's conjecture, form the Schubert basis for the cohomology of the loop Grassmannian. We derive several properties of these functions that extend those of skew Schur functions.

Abstract:
We obtain general identities for the product of two Schur functions in the case where one of the functions is indexed by a rectangular partition, and give their t-analogs using vertex operators. We study subspaces forming a filtration for the symmetric function space that lends itself to generalizing the theory of Schur functions and also provides a convenient environment for studying the Macdonald polynomials. We use our identities to prove that the vertex operators leave such subspaces invariant. We finish by showing that these operators act simply on the k-Schur functions, thus leading to a concept of irreducibility for these functions.

Abstract:
We consider a filtration of the symmetric function space given by $\Lambda^{(k)}_t$, the linear span of Hall-Littlewood polynomials indexed by partitions whose first part is not larger than $k$. We introduce symmetric functions called the $k$-Schur functions, providing an analog for the Schur functions in the subspaces $\Lambda^{(k)}_t$. We prove several properties for the $k$-Schur functions including that they form a basis for these subspaces that reduces to the Schur basis when $k$ is large. We also show that the connection coefficients for the $k$-Schur function basis with the Macdonald polynomials belonging to $\Lambda^{(k)}_t$ are polynomials in $q$ and $t$ with integral coefficients. In fact, we conjecture that these integral coefficients are actually positive, and give several other conjectures generalizing Schur function theory.

Abstract:
The Macdonald polynomials expanded in terms of a modified Schur function basis have coefficients called the $q,t$-Kostka polynomials. We define operators to build standard tableaux and show that they are equivalent to creation operators that recursively build the Macdonald polynomials indexed by two part partitions. We uncover a new basis for these particular Macdonald polynomials and in doing so are able to give an explicit description of their associated $q,t$-Kostka coefficients by assigning a statistic in $q$ and $t$ to each standard tableau.

Abstract:
Abnormal behaviors involving dopaminergic gene polymorphisms often reflect an insufficiency of usual feelings of satisfaction, or Reward Deficiency Syndrome (RDS). RDS results from a dysfunction in the “brain reward cascade,” a complex interaction among neurotransmitters (primarily dopaminergic and opioidergic). Individuals with a family history of alcoholism or other addictions may be born with a deficiency in the ability to produce or use these neurotransmitters. Exposure to prolonged periods of stress and alcohol or other substances also can lead to a corruption of the brain reward cascade function. We evaluated the potential association of four variants of dopaminergic candidate genes in RDS (dopamine D1 receptor gene [DRD1]; dopamine D2 receptor gene [DRD2]; dopamine transporter gene [DAT1]; dopamine beta-hydroxylase gene [DBH]). Methodology: We genotyped an experimental group of 55 subjects derived from up to five generations of two independent multiple-affected families compared to rigorously screened control subjects (e.g., N = 30 super controls for DRD2 gene polymorphisms). Data related to RDS behaviors were collected on these subjects plus 13 deceased family members. Results: Among the genotyped family members, the DRD2 Taq1 and the DAT1 10/10 alleles were significantly (at least p < 0.015) more often found in the RDS families vs. controls. The TaqA1 allele occurred in 100% of Family A individuals (N = 32) and 47.8% of Family B subjects (11 of 23). No significant differences were found between the experimental and control positive rates for the other variants. Conclusions: Although our sample size was limited, and linkage analysis is necessary, the results support the putative role of dopaminergic polymorphisms in RDS behaviors. This study shows the importance of a nonspecific RDS phenotype and informs an understanding of how evaluating single subset behaviors of RDS may lead to spurious results. Utilization of a nonspecific “reward” phenotype may be a paradigm shift in future association and linkage studies involving dopaminergic polymorphisms and other neurotransmitter gene candidates.

Abstract:
For the analysis of microRNA, a common approach is to first extract microRNA from cellular samples prior to any specific microRNA detection. Thus, it is important to determine the quality and yield of extracted microRNA. In this study, solid-phase extraction was used to isolate small RNA (<200 nt), which included microRNA, from mouse brain tissues. By using standard UV absorbance measurements, the amount of small RNA in the extracted RNA samples was determined. To determine the presence of microRNA, each RNA sample was analyzed by PAGE with SYBR^{?} Green II staining. Testing for contamination of any small DNA fragments, RNase and cellular peptides or proteins were systematically carried out. By scanning the gel image obtained from PAGE analysis, the average percentage of total microRNA (19 - 25 nt) in the extracted RNA samples was determined to be equal to 2.3 ± 0.5%. The yield of total microRNA was calculated to be ~0.5ng of microRNA per milligram of frozen mouse brain tissue. In comparison to other methods that require the use of expensive specialized instrumentation, the approach of combining the standard UV absorbance and PAGE analysis represents a simple and viable method for evaluating the quality and yield of microRNA extraction from tissue samples.

Abstract:
Let $\Lambda$ be the space of symmetric functions and $V_k$ be the subspace spanned by the modified Schur functions $\{S_\lambda[X/(1-t)]\}_{\lambda_1\leq k}$. We introduce a new family of symmetric polynomials, $\{A_{\lambda}^{(k)}[X;t]\}_{\lambda_1\leq k}$, constructed from sums of tableaux using the charge statistic. We conjecture that the polynomials $A_{\lambda}^{(k)}[X;t]$ form a basis for $V_k$ and that the Macdonald polynomials indexed by partitions whose first part is not larger than $k$ expand positively in terms of our polynomials. A proof of this conjecture would not only imply the Macdonald positivity conjecture, but would substantially refine it. Our construction of the $A_\lambda^{(k)}[X;t]$ relies on the use of tableaux combinatorics and yields various properties and conjectures on the nature of these polynomials. Another important development following from our investigation is that the $A_{\lambda}^{(k)}[X;t]$ seem to play the same role for $V_k$ as the Schur functions do for $\Lambda$. In particular, this has led us to the discovery of many generalizations of properties held by the Schur functions, such as Pieri and Littlewood-Richardson type coefficients.

Abstract:
We show that the action of classical operators associated to the Macdonald polynomials on the basis of Schur functions, S_{\lambda}[X(t-1)/(q-1)], can be reduced to addition in \lambda-rings. This provides explicit formulas for the Macdonald polynomials expanded in this basis as well as in the ordinary Schur basis, S_{\lambda}[X], and the monomial basis, m_{\lambda}[X].