An Elementary Proof of Dodgson's Condensation Method for Calculating Determinants

In 1866, Charles Ludwidge Dodgson published a paper concerning a method for evaluating determinants called the condensation method. His paper documented a new method to calculate determinants that was based on Jacobi's Theorem. The condensation method is presented and proven here, and is demonstrated by a series of examples. The condensation method can be applied to a number of situations, including calculating eigenvalues, solving a system of linear equations, and even determining the different energy levels of a molecular system. The method is much more efficient than cofactor expansions, particularly for large matrices; for a 5 x 5 matrix, the condensation method requires about half as many calculations. Zeros appearing in the interior of a matrix can cause problems, but a way around the issue can usually be found. Overall, Dodgson's condensation method is an interesting and simple way to find determinants. This paper presents an elementary proof of Dodgson's method.Comment: 7 pages, no figure

Exceptional Points in Atomic Spectra

We report the existence of exceptional points for the hydrogen atom in crossed magnetic and electric fields in numerical calculations. The resonances of the system are investigated and it is shown how exceptional points can be found by exploiting characteristic properties of the degeneracies, which are branch point singularities. A possibility for the observation of exceptional points in an experiment with atoms is proposed.Comment: 4 pages, 4 figures, 1 table, to be published in Physical Review Letter

Sound propagation over uneven ground and irregular topography

The development of theoretical, computational, and experimental techniques for predicting the effects of irregular topography on long range sound propagation in the atmosphere is discussed. Irregular topography here is understood to imply a ground surface that (1) is not idealizable as being perfectly flat or (2) that is not idealizable as having a constant specific acoustic impedance. The study focuses on circumstances where the propagation is similar to what might be expected for noise from low-altitude air vehicles flying over suburban or rural terrain, such that rays from the source arrive at angles close to grazing incidence

Reduced healthcare utilisation following successful HCV treatment in HIV co-infected patients with mild liver disease

New direct-acting antivirals (DAA) for hepatitis C virus (HCV) infection have achieved high cure rates in many patient groups previously considered difficult-to-treat, including those HIV/HCV co-infected. The high price of these medications is likely to limit access to treatment, at least in the short term. Early treatment priority is likely to be given to those with advanced disease, but a more detailed understanding of the potential benefits in treating those with mild disease is needed. We hypothesized that successful HCV treatment within a co-infected population with mild liver disease would lead to a reduction in the use and costs of healthcare services in the 5 years following treatment completion. We performed a retrospective cohort study of HIV/HCV-co-infected patients without evidence of fibrosis/cirrhosis who received a course of HCV therapy between 2004 and 2013. Detailed analysis of healthcare utilization up to 5 years following treatment for each patient using clinical and electronic records was used to estimate healthcare costs. Sixty-three patients were investigated, of whom 48 of 63 (76.2%) achieved sustained virological response 12 weeks following completion of therapy (SVR12). Individuals achieving SVR12 incurred lower health utilization costs (£5000 per-patient) compared to (£10 775 per-patient) non-SVR patients in the 5 years after treatment. Healthcare utilization rates and costs in the immediate 5 years following treatment were significantly higher in co-infected patients with mild disease that failed to achieve SVR12. These data suggest additional value to achieving cure beyond the prevention of complications of disease

Bose-Einstein condensates with attractive 1/r interaction: The case of self-trapping

Amplifying on a proposal by O'Dell et al. for the realization of Bose-Einstein condensates of neutral atoms with attractive $1/r$ interaction, we point out that the instance of self-trapping of the condensate, without external trap potential, is physically best understood by introducing appropriate "atomic" units. This reveals a remarkable scaling property: the physics of the condensate depends only on the two parameters $N^2 a/a_u$ and $\gamma/N^2$, where $N$ is the particle number, $a$ the scattering length, $a_u$ the "Bohr" radius and $\gamma$ the trap frequency in atomic units. We calculate accurate numerical results for self-trapping wave functions and potentials, for energies, sizes and peak densities, and compare with previous variational results. As a novel feature we point out the existence of a second solution of the extended Gross-Pitaevskii equation for negative scattering lengths, with and without trapping potential, which is born together with the ground state in a tangent bifurcation. This indicates the existence of an unstable collectively excited state of the condensate for negative scattering lengths.Comment: 7 pages, 7 figures, to appear in Phys. Rev.

A Minimal Periods Algorithm with Applications

Kosaraju in ``Computation of squares in a string'' briefly described a linear-time algorithm for computing the minimal squares starting at each position in a word. Using the same construction of suffix trees, we generalize his result and describe in detail how to compute in O(k|w|)-time the minimal k-th power, with period of length larger than s, starting at each position in a word w for arbitrary exponent $k\geq2$ and integer $s\geq0$. We provide the complete proof of correctness of the algorithm, which is somehow not completely clear in Kosaraju's original paper. The algorithm can be used as a sub-routine to detect certain types of pseudo-patterns in words, which is our original intention to study the generalization.Comment: 14 page

Longest Common Extensions in Sublinear Space

The longest common extension problem (LCE problem) is to construct a data structure for an input string $T$ of length $n$ that supports LCE$(i,j)$ queries. Such a query returns the length of the longest common prefix of the suffixes starting at positions $i$ and $j$ in $T$. This classic problem has a well-known solution that uses $O(n)$ space and $O(1)$ query time. In this paper we show that for any trade-off parameter $1 \leq \tau \leq n$, the problem can be solved in $O(\frac{n}{\tau})$ space and $O(\tau)$ query time. This significantly improves the previously best known time-space trade-offs, and almost matches the best known time-space product lower bound.Comment: An extended abstract of this paper has been accepted to CPM 201