Low-energy molecular collisions in a permanent magnetic trap

Cold, neutral hydroxyl radicals are Stark decelerated and confined within a magnetic trap consisting of two permanent ring magnets. The OH molecules are trapped in the ro-vibrational ground state at a density of $\sim10^{6}$ cm$^{-3}$ and temperature of 70 mK. Collisions between the trapped OH sample and supersonic beams of atomic He and molecular D$_{2}$ are observed and absolute collision cross sections measured. The He--OH and D$_{2}$--OH center-of-mass collision energies are tuned from 60 cm$^{-1}$ to 230 cm$^{-1}$ and 145 cm$^{-1}$ to 510 cm$^{-1}$, respectively, yielding evidence of reduced He--OH inelastic cross sections at energies below 84 cm$^{-1}$, the OH ground rotational level spacing.Comment: 4 pages, 4 figure

Amino Acids and Peptides. XIV, Synthesis of a Tetrapeptide Sequence (A5-A8) of Glucagon

Author Institution: Department of Chemistry, Stanford University, Stanford, California 94305A synthesis of the tetrapeptide sequence A5-A8 of glucagon is described that employs various blocking groups, coupling procedures, and routes

Benford Behavior of Zeckendorf Decompositions

A beautiful theorem of Zeckendorf states that every integer can be written uniquely as the sum of non-consecutive Fibonacci numbers $\{ F_i \}_{i = 1}^{\infty}$. A set $S \subset \mathbb{Z}$ is said to satisfy Benford's law if the density of the elements in $S$ with leading digit $d$ is $\log_{10}{(1+\frac{1}{d})}$; in other words, smaller leading digits are more likely to occur. We prove that, as $n\to\infty$, for a randomly selected integer $m$ in $[0, F_{n+1})$ the distribution of the leading digits of the Fibonacci summands in its Zeckendorf decomposition converge to Benford's law almost surely. Our results hold more generally, and instead of looking at the distribution of leading digits one obtains similar theorems concerning how often values in sets with density are attained.Comment: Version 1.0, 12 pages, 1 figur

Gaussian Behavior of the Number of Summands in Zeckendorf Decompositions in Small Intervals

Zeckendorf's theorem states that every positive integer can be written uniquely as a sum of non-consecutive Fibonacci numbers ${F_n}$, with initial terms $F_1 = 1, F_2 = 2$. We consider the distribution of the number of summands involved in such decompositions. Previous work proved that as $n \to \infty$ the distribution of the number of summands in the Zeckendorf decompositions of $m \in [F_n, F_{n+1})$, appropriately normalized, converges to the standard normal. The proofs crucially used the fact that all integers in $[F_n, F_{n+1})$ share the same potential summands. We generalize these results to subintervals of $[F_n, F_{n+1})$ as $n \to \infty$; the analysis is significantly more involved here as different integers have different sets of potential summands. Explicitly, fix an integer sequence $\alpha(n) \to \infty$. As $n \to \infty$, for almost all $m \in [F_n, F_{n+1})$ the distribution of the number of summands in the Zeckendorf decompositions of integers in the subintervals $[m, m + F_{\alpha(n)})$, appropriately normalized, converges to the standard normal. The proof follows by showing that, with probability tending to $1$, $m$ has at least one appropriately located large gap between indices in its decomposition. We then use a correspondence between this interval and $[0, F_{\alpha(n)})$ to obtain the result, since the summands are known to have Gaussian behavior in the latter interval. % We also prove the same result for more general linear recurrences.Comment: Version 1.0, 8 page

Magneto-electrostatic trapping of ground state OH molecules

We report the magnetic confinement of neutral, ground state hydroxyl radicals (OH) at a density of $\sim3\times10^{3}$ cm$^{-3}$ and temperature of $\sim$30 mK. An adjustable electric field of sufficient magnitude to polarize the OH is superimposed on the trap in either a quadrupole or homogenous field geometry. The OH is confined by an overall potential established via molecular state mixing induced by the combined electric and magnetic fields acting on the molecule's electric dipole and magnetic dipole moments, respectively. An effective molecular Hamiltonian including Stark and Zeeman terms has been constructed to describe single molecule dynamics inside the trap. Monte Carlo simulation using this Hamiltonian accurately models the observed trap dynamics in various trap configurations. Confinement of cold polar molecules in a magnetic trap, leaving large, adjustable electric fields for control, is an important step towards the study of low energy dipole-dipole collisions.Comment: 4 pages, 4 figure

OH hyperfine ground state: from precision measurement to molecular qubits

We perform precision microwave spectroscopy--aided by Stark deceleration--to reveal the low magnetic field behavior of OH in its ^2\Pi_{3/2} ro-vibronic ground state, identifying two field-insensitive hyperfine transitions suitable as qubits and determining a differential Lande g-factor of 1.267(5)\times10^{-3} between opposite parity components of the \Lambda-doublet. The data are successfully modeled with an effective hyperfine Zeeman Hamiltonian, which we use to make a tenfold improvement of the magnetically sensitive, astrophysically important \Delta F=\pm1 satellite-line frequencies, yielding 1720529887(10) Hz and 1612230825(15) Hz.Comment: 4+ pages, 3 figure

Pay-As-You-Go Driving: Examining Possible Road-User Charge Rate Structures for California

This report lays out principles to help California policymakers identify an optimal rate structure for a road-user charge (RUC). The rate structure is different from the rate itself. The rate is the price a driver pays, while the structure is the set of principles that govern how that price is set. We drew on existing research on rate setting in transportation, public utilities, and behavioral economics to develop a set of conceptual principles that can be used to evaluate rate structures, and then applied these principles to a set of mileage fee rate structure options. Key findings include that transportation system users already pay for driving using a wide array of rate structures, including some that charge rate structured based on vehicle characteristics, user characteristics, and time or location of driving. We also conclude that the principal advantage of RUCs is not their ability to raise revenue but rather to variably allocate charges among various types of users and travelers. To obtain those benefits, policymakers need to proactively design rate structures to advance important state policy goals and/or improve administrative and political feasibility

Charging Drivers by the Gallon vs. the Mile: An Equity Analysis by Geography and Income in California

This study used data from the 2017 National Household Travel Survey California Add-On sample to explore how replacing the current state vehicle fuel tax with a flat-per-mile-rate road-user charge (RUC) would affect costs for different kinds of households. We first estimated how household vehicle fuel efficiency, mileage, and fuel tax expenditures vary by geography (rural vs. urban) and by income. These findings were then used to estimate how much different types of households pay in the current per-gallon state fuel tax, what they would pay if the state were to replace fuel taxes with a flat-rate road-usage charge (RUC) that would generate revenues similar to the current state fuel tax (2.52¢ per mile driven), and the difference in household expenditures between the fuel tax and RUC. We find that rural households tend to drive more miles and own less fuel-efficient vehicles than urban ones, so they pay comparatively more in fuel tax and would pay more with the RUC as well. However, this rural/urban variation is less for the RUC than the fuel tax, so moving to a flat-rate RUC would redistribute some of the overall tax burden from rural households (that drive more miles in fuel-thirsty vehicles) to urban households (that drive fewer miles in more fuel-efficient vehicles). Transitioning from the fuel tax to RUC would also generally shift the fuel tax burden from lower-income to higher-income households, with one exception: expenditures would rise for low-income urban households. However, the variation in the tax incidence between the gas tax and RUC is quite modest, amounting to less than one dollar per week for both urban and rural households at all income levels