The Schrodinger-like Equation for a Nonrelativistic Electron in a Photon Field of Arbitrary Intensity

The ordinary Schrodinger equation with minimal coupling for a nonrelativistic electron interacting with a single-mode photon field is not satisfied by the nonrelativistic limit of the exact solutions to the corresponding Dirac equation. A Schrodinger-like equation valid for arbitrary photon intensity is derived from the Dirac equation without the weak-field assumption. The "eigenvalue" in the new equation is an operator in a Cartan subalgebra. An approximation consistent with the nonrelativistic energy level derived from its relativistic value replaces the "eigenvalue" operator by an ordinary number, recovering the ordinary Schrodinger eigenvalue equation used in the formal scattering formalism. The Schrodinger-like equation for the multimode case is also presented.Comment: Tex file, 13 pages, no figur

On the least common multiple of qq-binomial coefficients

In this paper, we prove the following identity \lcm({n\brack 0}_q,{n\brack 1}_q,...,{n\brack n}_q) =\frac{\lcm([1]_q,[2]_q,...,[n+1]_q)}{[n+1]_q}, where ${n\brack k}_q$ denotes the $q$-binomial coefficient and $[n]_q=\frac{1-q^n}{1-q}$. This result is a $q$-analogue of an identity of Farhi [Amer. Math. Monthly, November (2009)].Comment: 5 page

Visualisation of Origins, Destinations and Flows with OD Maps

We present a new technique for the visual exploration of origins (O) and destinations (D) arranged in geographic space. Previous attempts to map the flows between origins and destinations have suffered from problems of occlusion usually requiring some form of generalisation, such as aggregation or flow density estimation before they can be visualized. This can lead to loss of detail or the introduction of arbitrary artefacts in the visual representation. Here, we propose mapping OD vectors as cells rather than lines, comparable with the process of constructing OD matrices, but unlike the OD matrix, we preserve the spatial layout of all origin and destination locations by constructing a gridded two‐level spatial treemap. The result is a set of spatially ordered small multiples upon which any arbitrary geographic data may be projected. Using a hash grid spatial data structure, we explore the characteristics of the technique through a software prototype that allows interactive query and visualisation of 105‐106 simulated and recorded OD vectors. The technique is illustrated using US county to county migration and commuting statistics

Factors of sums and alternating sums involving binomial coefficients and powers of integers

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers $n_1,..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer $r$, there holds {align*} \sum_{k=0}^{n_1}\epsilon^k (2k+1)^{2r+1}\prod_{i=1}^{m} {n_i+n_{i+1}+1\choose n_i-k} \equiv 0 \mod (n_1+n_m+1){n_1+n_m\choose n_1}, {align*} and conjecture that for any nonnegative integer $r$ and positive integer $s$ such that $r+s$ is odd, $$\sum_{k=0}^{n}\epsilon ^k (2k+1)^{r}({2n\choose n-k}-{2n\choose n-k-1})^{s} \equiv 0 \mod{{2n\choose n}},$$ where $\epsilon=\pm 1$.Comment: 14 pages, to appear in Int. J. Number Theor

Metastable helium molecules as tracers in superfluid liquid 4^{4}He

Metastable helium molecules generated in a discharge near a sharp tungsten tip operated in either pulsed mode or continuous field-emission mode in superfluid liquid $^{4}$He are imaged using a laser-induced-fluorescence technique. By pulsing the tip, a small cloud of He$_{2}^{*}$ molecules is produced. At 2.0 K, the molecules in the liquid follow the motion of the normal fluid. We can determine the normal-fluid velocity in a heat-induced counterflow by tracing the position of a single molecule cloud. As we run the tip in continuous field-emission mode, a normal-fluid jet from the tip is generated and molecules are entrained in the jet. A focused 910 nm pump laser pulse is used to drive a small group of molecules to the vibrational $a(1)$ state. Subsequent imaging of the tagged $a(1)$ molecules with an expanded 925 nm probe laser pulse allows us to measure the velocity of the normal fluid. The techniques we developed demonstrate for the first time the ability to trace the normal-fluid component in superfluid helium using angstrom-sized particles.Comment: 4 pages, 7 figures. Submitted to Phys. Rev. Let