Notes on highest weight modules of the elliptic algebra Aq,p(sl^2){\cal A}_{q,p}\left(\widehat{sl}_2\right)

We discuss a construction of highest weight modules for the recently defined elliptic algebra ${\cal A}_{q,p}(\widehat{sl}_2)$, and make several conjectures concerning them. The modules are generated by the action of the components of the operator $L$ on the highest weight vectors. We introduce the vertex operators $\Phi$ and $\Psi^*$ through their commutation relations with the $L$-operator. We present ordering rules for the $L$- and $\Phi$-operators and find an upper bound for the number of linearly independent vectors generated by them, which agrees with the known characters of $\widehat{sl}_2$-modules.Comment: Nonstandard macro package eliminate

Combined congenic mapping and nuclease-based gene targeting for studying allele-specific effects of Tnfrsf9 within the Idd9.3 autoimmune diabetes locus.

Rodent complex trait genetic studies involving a cross between two inbred strains are usually followed by congenic mapping to refine the loci responsible for the phenotype. However, progressing from a chromosomal region to the actual causal gene remains challenging because multiple polymorphic genes are often closely linked. The goal of this study was to develop a strategy that allows candidate gene testing by allele-specific expression without prior knowledge of the credible causal variant. Tnfrsf9 (encoding CD137) is a candidate gene for the Idd9.3 type 1 diabetes (T1D) susceptibility locus in the nonobese diabetic (NOD) mouse model. A C57BL/10Sn (B10)-derived diabetes resistance Idd9.3 congenic region has been shown to enhance accumulation of CD137+ regulatory T cells and serum soluble CD137 in NOD mice. By combining the power of congenic mapping and nuclease-based gene targeting, we established a system where a pair of F1 hybrids expressed either the B10 or NOD Tnfrsf9 allele mimicking coisogenic strains. Using this approach, we demonstrated that the allelic difference in B10 and NOD Tnfrsf9 alone was sufficient to cause differential accumulation of CD137+ regulatory T cells and serum soluble CD137 levels. This strategy can be broadly applied to other rodent genetic mapping studies

Infinite Hopf family of elliptic algebras and bosonization

Elliptic current algebras E_{q,p}(\hat{g}) for arbitrary simply laced finite dimensional Lie algebra g are defined and their co-algebraic structures are studied. It is shown that under the Drinfeld like comultiplications, the algebra E_{q,p}(\hat{g}) is not co-closed for any g. However putting the algebras E_{q,p}(\hat{g}) with different deformation parameters together, we can establish a structure of infinite Hopf family of algebras. The level 1 bosonic realization for the algebra E_{q,p}(\hat{g}) is also established.Comment: LaTeX, 11 pages. This is the new and final versio

Evaluation of plasma neurotransmitters in children living with Attention-Deficit Hyperactivity Disorder (ADHD)

This study aimed to ascertain the underlying neuro-biochemical imbalances that exist in children with ADHD by assessing the plasma levels of dopamine, serotonin, norepinephrine, γ aminobutyric acid (GABA) and glutamate. Moreover, it investigated the potential effects of PUFA and vitamins supplementation as an alternative therapy to modulate the levels of these neurotransmitters and the overall clinical status of ADHD patients. The study included 40 ADHD patients, aged 4-6 years. The diagnostic and statistical manual of mental disorders (DSM-5) test has been employed to diagnose patients with ADHD and the severity of symptoms was assessed using the Arabic version of Conners' Parent Rating Scale. Additionally, patients were assessed using the Arabic versions of Mini International Neuropsychiatric Interview for Children (M.I.N.I. Kid) and Stanford-Binet Intelligence Scale, 5th Edition (SB5). Recruited patient received nutritional supplement of semi-solid diet containing 1000 mg PUFA with selected vitamins once daily for six months. The evaluation of ADHD symptoms and levels of neurotransmitters has been carried out at pre-/post-intervention stage. Post-nutrition intervention assessment, there was a significant increase in dopamine, norepinephrine, GABA levels (p-value < 0.0001) with significant decrease in glutamate level (p-value < 0.0001) when compared to their correspondent pre-intervention levels. Symptoms of inattention and/or hyperactivity-impulsivity were significantly improved after 6 months nutrition intervention program (p-value<0.001). Therefore, supplementation with omega-3 fatty acids and vitamins could be considered more extensively in therapy of ADHD patients particularly those who are less than 6 years old

Mixing of Ground States in Vertex Models

We consider the analogue of the 6-vertex model constructed from alternating spin n/2 and spin m/2 lines, where $1\leq n<m$. We identify the transfer matrix and the space on which it acts in terms of the representation theory of $U_q(sl_2)$. We diagonalise the transfer matrix and compute the S-matrix. We give a trace formula for local correlation functions. When n=1, the 1-point function of a spin m/2 local variable for the alternating lattice with a particular ground state is given as a linear combination of the 1-point functions of the pure spin m/2 model with different ground states. The mixing ratios are calculated exactly and are expressed in terms of irreducible characters of $U_q(sl_2)$ and the deformed Virasoro algebra.Comment: 12 pages, LaTeX, typos correcte

Diagonalization of the XXZ Hamiltonian by Vertex Operators

We diagonalize the anti-ferroelectric XXZ-Hamiltonian directly in the thermodynamic limit, where the model becomes invariant under the action of affine U_q( sl(2) ). Our method is based on the representation theory of quantum affine algebras, the related vertex operators and KZ equation, and thereby bypasses the usual process of starting from a finite lattice, taking the thermodynamic limit and filling the Dirac sea. From recent results on the algebraic structure of the corner transfer matrix of the model, we obtain the vacuum vector of the Hamiltonian. The rest of the eigenvectors are obtained by applying the vertex operators, which act as particle creation operators in the space of eigenvectors. We check the agreement of our results with those obtained using the Bethe Ansatz in a number of cases, and with others obtained in the scaling limit --- the $su(2)$-invariant Thirring model.Comment: 65 page

(l,q)(l,q)-Deformed Grassmann Field and the Two-dimensional Ising Model

In this paper we construct the exact representation of the Ising partition function in the form of the $SL_q(2,R)$-invariant functional integral for the lattice free $(l,q)$-fermion field theory ($l=q=-1$). It is shown that the $(l,q)$-fermionization allows one to re-express the partition function of the eight-vertex model in external field through functional integral with four-fermion interaction. To construct these representations, we define a lattice $(l,q,s)$-deformed Grassmann bispinor field and extend the Berezin integration rules to this field. At $l=q=-1, s=1$ we obtain the lattice $(l,q)$-fermion field which allows us to fermionize the two-dimensional Ising model. We show that the Gaussian integral over $(q,s)$-Grassmann variables is expressed through the $(q,s)$-deformed Pfaffian which is equal to square root of the determinant of some matrix at $q=\pm 1, s=\pm 1$.Comment: 24 pages, LaTeX; minor change

A generalization of the q-Saalschutz sum and the Burge transform

A generalization of the q-(Pfaff)-Saalschutz summation formula is proved. This implies a generalization of the Burge transform, resulting in an additional dimension of the ``Burge tree''. Limiting cases of our summation formula imply the (higher-level) Bailey lemma, provide a new decomposition of the q-multinomial coefficients, and can be used to prove the Lepowsky and Primc formula for the A_1^{(1)} string functions.Comment: 18 pages, AMSLaTe

On two-point boundary correlations in the six-vertex model with DWBC

The six-vertex model with domain wall boundary conditions (DWBC) on an N x N square lattice is considered. The two-point correlation function describing the probability of having two vertices in a given state at opposite (top and bottom) boundaries of the lattice is calculated. It is shown that this two-point boundary correlator is expressible in a very simple way in terms of the one-point boundary correlators of the model on N x N and (N-1) x (N-1) lattices. In alternating sign matrix (ASM) language this result implies that the doubly refined x-enumerations of ASMs are just appropriate combinations of the singly refined ones.Comment: v2: a reference added, typos correcte