Invariable generation of the symmetric group

We say that permutations $\pi_1,\dots, \pi_r \in \mathcal{S}_n$ invariably generate $\mathcal{S}_n$ if, no matter how one chooses conjugates $\pi'_1,\dots,\pi'_r$ of these permutations, $\pi'_1,\dots,\pi'_r$ generate $\mathcal{S}_n$. We show that if $\pi_1,\pi_2,\pi_3$ are chosen randomly from $\mathcal{S}_n$ then, with probability tending to 1 as $n \rightarrow \infty$, they do not invariably generate $\mathcal{S}_n$. By contrast it was shown recently by Pemantle, Peres and Rivin that four random elements do invariably generate $\mathcal{S}_n$ with positive probability. We include a proof of this statement which, while sharing many features with their argument, is short and completely combinatorial.Comment: 15 pages. Corrections and clarifications suggested by the referees. Added a reference to a paper of Raouj and Stef which solves related problems about divisor

Large gaps between consecutive prime numbers

Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a question of Erdos, we show that $$G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2},$$ where $f(X)$ is a function tending to infinity with $X$. Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the existence and distribution of long arithmetic progressions consisting entirely of primes.Comment: v2. very minor corrections. To appear in Ann. Mat

Attacking Without Provocation; Maintaining a Subsequent Occupation; All for the Sake of Democratization: The Constitutionality and Wisdom of Preventive War and Post-Conflict Nation-Building

This Note examines the constitutionality and wisdom of both aspects of the current warfare policy. Ultimately, the current United States warfare strategy that includes using preventive wars followed by post-conflict military nation-building is constitutional; however, underlying policy concerns suggest that this approach to warfare should be reconsidered and reframed

Unfair, Abusive, and Unlawful: Protecting Debtors and Society from Unrestrained Bank Account Garnishment

In the span of a generation, consumer credit has reshaped the financial lives of millions of Americans. Today, some seventy million Americans have a debt in collections, and creditors file millions of actions annually to secure repayment of these loans. Despite the rapid expansion of consumer debt, the Consumer Credit Protection Act, the only federal law limiting garnishment, has not been updated since its enactment in 1968. Moreover, courts have narrowly construed its provisions to permit creditors to empty a debtor’s bank account to repay a delinquent debt. To afford debtors the basic protections of the Consumer Credit Protection Act, this Note argues that courts should enforce its provisions to protect wages deposited into a bank account. Alternatively, considering the dramatic expansion of consumer credit and debt collection, this Note argues that the Consumer Financial Protection Bureau should engage in rulemaking to prohibit the unrestrained garnishment of bank accounts as an unfair and abusive practice. These federal interventions are critical in shielding debtors and their families from destitution and in protecting the public from the onerous social costs of unrestrained debt collection

Total Error and Variability Measures with Integrated Disclosure Limitation for Quarterly Workforce Indicators and LEHD Origin Destination Employment Statistics in OnThe Map

We report results from the first comprehensive total quality evaluation of five major indicators in the U.S. Census Bureau\u27s Longitudinal Employer-Household Dynamics (LEHD) Program Quarterly Workforce Indicators (QWI): total employment, beginning-of-quarter employment, full-quarter employment, total payroll, and average monthly earnings of full-quarter employees. Beginning-of-quarter employment is also the main tabulation variable in the LEHD Origin-Destination Employment Statistics (LODES) workplace reports as displayed in OnTheMap (OTM). The evaluation is conducted by generating multiple threads of the edit and imputation models used in the LEHD Infrastructure File System. These threads conform to the Rubin (1987) multiple imputation model, with each thread or implicate being the output of formal probability models that address coverage, edit, and imputation errors. Design-based sampling variability and finite population corrections are also included in the evaluation. We derive special formulas for the Rubin total variability and its components that are consistent with the disclosure avoidance system used for QWI and LODES/OTM workplace reports. These formulas allow us to publish the complete set of detailed total quality measures for QWI and LODES. The analysis reveals that the five publication variables under study are estimated very accurately for tabulations involving at least 10 jobs. Tabulations involving three to nine jobs have quality in the range generally deemed acceptable. Tabulations involving zero, one or two jobs, which are generally suppressed in the QWI and synthesized in LODES, have substantial total variability but their publication in LODES allows the formation of larger custom aggregations, which will in general have the accuracy estimated for tabulations in the QWI based on a similar number of workers

Lower Bounds for Oblivious Near-Neighbor Search

We prove an $\Omega(d \lg n/ (\lg\lg n)^2)$ lower bound on the dynamic cell-probe complexity of statistically $\mathit{oblivious}$ approximate-near-neighbor search ($\mathsf{ANN}$) over the $d$-dimensional Hamming cube. For the natural setting of $d = \Theta(\log n)$, our result implies an $\tilde{\Omega}(\lg^2 n)$ lower bound, which is a quadratic improvement over the highest (non-oblivious) cell-probe lower bound for $\mathsf{ANN}$. This is the first super-logarithmic $\mathit{unconditional}$ lower bound for $\mathsf{ANN}$ against general (non black-box) data structures. We also show that any oblivious $\mathit{static}$ data structure for decomposable search problems (like $\mathsf{ANN}$) can be obliviously dynamized with $O(\log n)$ overhead in update and query time, strengthening a classic result of Bentley and Saxe (Algorithmica, 1980).Comment: 28 page

Stretching the Twenty-First Amendment Beyond Its Limit: A Critique of S. Wine & Spirits of Am., Inc. v. Div. of Alcohol & Tobacco Control

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A prototype system for detecting the radio-frequency pulse associated with cosmic ray air showers

The development of a system to detect the radio-frequency (RF) pulse associated with extensive air showers of cosmic rays is described. This work was performed at the CASA/MIA array in Utah, with the intention of designing equipment that can be used in conjunction with the Auger Giant Array. A small subset of data (less than 40 out of a total of 600 hours of running time), taken under low-noise conditions, permitted upper limits to be placed on the rate for pulses accompanying showers of energies around $10^{17}$ eV.Comment: 53 pages, LaTeX, 19 figures, published in Nuclear Instruments and Methods. Revised version; some references update