Eigenproblem for Jacobi matrices: hypergeometric series solution

We study the perturbative power-series expansions of the eigenvalues and eigenvectors of a general tridiagonal (Jacobi) matrix of dimension d. The(small) expansion parameters are being the entries of the two diagonals of length d-1 sandwiching the principal diagonal, which gives the unperturbed spectrum. The solution is found explicitly in terms of multivariable (Horn-type) hypergeometric series of 3d-5 variables in the generic case, or 2d-3 variables for the eigenvalue growing from a corner matrix element. To derive the result, we first rewrite the spectral problem for a Jacobi matrix as an equivalent system of cubic equations, which are then resolved by the application of the multivariable Lagrange inversion formula. The corresponding Jacobi determinant is calculated explicitly. Explicit formulae are also found for any monomial composed of eigenvector's components.Comment: Latex, 20 pages; v2: corrected typos, added section with example

A Super-Flag Landau Model

We consider the quantum mechanics of a particle on the coset superspace $SU(2|1)/[U(1)\times U(1)]$, which is a super-flag manifold with $SU(2)/U(1)\cong S^2$ `body'. By incorporating the Wess-Zumino terms associated with the $U(1)\times U(1)$ stability group, we obtain an exactly solvable super-generalization of the Landau model for a charged particle on the sphere. We solve this model using the factorization method. Remarkably, the physical Hilbert space is finite-dimensional because the number of admissible Landau levels is bounded by a combination of the U(1) charges. The level saturating the bound has a wavefunction in a shortened, degenerate, irrep of $SU(2|1)$

Q-operator and factorised separation chain for Jack polynomials

Applying Baxter's method of the Q-operator to the set of Sekiguchi's commuting partial differential operators we show that Jack polynomials P(x_1,...,x_n) are eigenfunctions of a one-parameter family of integral operators Q_z. The operators Q_z are expressed in terms of the Dirichlet-Liouville n-dimensional beta integral. From a composition of n operators Q_{z_k} we construct an integral operator S_n factorising Jack polynomials into products of hypergeometric polynomials of one variable. The operator S_n admits a factorisation described in terms of restricted Jack polynomials P(x_1,...,x_k,1,...,1). Using the operator Q_z for z=0 we give a simple derivation of a previously known integral representation for Jack polynomials.Comment: 26 page

Adenoid cystic carcinoma: emerging role of translocations and gene fusions.

Adenoid cystic carcinoma (ACC), the second most common salivary gland malignancy, is notorious for poor prognosis, which reflects the propensity of ACC to progress to clinically advanced metastatic disease. Due to high long-term mortality and lack of effective systemic treatment, the slow-growing but aggressive ACC poses a particular challenge in head and neck oncology. Despite the advancements in cancer genomics, up until recently relatively few genetic alterations critical to the ACC development have been recognized. Although the specific chromosomal translocations resulting in MYB-NFIB fusions provide insight into the ACC pathogenesis and represent attractive diagnostic and therapeutic targets, their clinical significance is unclear, and a substantial subset of ACCs do not harbor the MYB-NFIB translocation. Strategies based on detection of newly described genetic events (such as MYB activating super-enhancer translocations and alterations affecting another member of MYB transcription factor family-MYBL1) offer new hope for improved risk assessment, therapeutic intervention and tumor surveillance. However, the impact of these approaches is still limited by an incomplete understanding of the ACC biology, and the manner by which these alterations initiate and drive ACC remains to be delineated. This manuscript summarizes the current status of gene fusions and other driver genetic alterations in ACC pathogenesis and discusses new therapeutic strategies stemming from the current research

Pulsed Adiabatic Photoassociation via Scattering Resonances

We develop the theory for the Adiabatic Raman Photoassociation (ARPA) of ultracold atoms to form ultracold molecules in the presence of scattering resonances. Based on a computational method in which we replace the continuum with a discrete set of "effective modes", we show that the existence of resonances greatly aids in the formation of deeply bound molecular states. We illustrate our general theory by computationally studying the formation of $^{85}$Rb$_2$ molecules from pairs of colliding ultracold $^{85}$Rb atoms. The single-event transfer yield is shown to have a near-unity value for wide resonances, while the ensemble-averaged transfer yield is shown to be higher for narrow resonances. The ARPA yields are compared with that of (the experimentally measured) "Feshbach molecule" magneto-association. Our findings suggest that an experimental investigation of ARPA at sub-$\mu$K temperatures is warranted.Comment: 20 pages, 11 figure

Non-functional biomimicry : utilising natural patterns to provoke attention responses

Natural reoccurring patterns arise from chaos and are prevalent throughout nature. The formation of these patterns is controlled by, or produces, underlying geometrical structures. Biomimicry is the study of nature’s structure, processes and systems, as models and solutions for design challenges and is being widely utilized in order to address many issues of contemporary engineering. Many academics now believe that aesthetics stem from pattern recognition, consequently, aesthetic preference may be a result of individuals recognising, and interacting with, natural patterns. The goal of this research was to investigate the impact of specific naturally occurring pattern types (spiral, branching, and fractal patterns) on user behaviour; investigating the potential of such patterns to control and influence how individuals interact with their surrounding environment. The results showed that the underlying geometry of natural patterns has the potential to induce attention responses to a statistically significant level