2,200 research outputs found
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
, which is a super-flag manifold with
`body'. By incorporating the Wess-Zumino terms associated
with the 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
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
Rb molecules from pairs of colliding ultracold 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-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
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