Counting fixed points and rooted closed walks of the singular map x↦xxnx \mapsto x^{x^n} modulo powers of a prime

The "self-power" map $x \mapsto x^x$ modulo $m$ and its generalized form $x \mapsto x^{x^n}$ modulo $m$ are of considerable interest for both theoretical reasons and for potential applications to cryptography. In this paper, we use $p$-adic methods, primarily $p$-adic interpolation, Hensel's lemma, and lifting singular points modulo $p$, to count fixed points and rooted closed walks of equations related to these maps when $m$ is a prime power. In particular, we introduce a new technique for lifting singular solutions of several congruences in several unknowns using the left kernel of the Jacobian matrix.Comment: 18 pages. Version 2 shortens proofs, reduces redundancy, and introduces new technique for counting rooted closed walks. Version 3 updates title to agree with journal publicatio

The Igusa Local Zeta Function Associated with the Singular Cases of the Determinant and the Pfaffian

AbstractThis paper describes the theory of the Igusa local zeta function associated with a polynomialf(x) with coefficients in ap-adic local fieldK. Results are given in two cases wheref(x) is the determinant of a Hermitian matrix of degreemwith coefficients in: (1) a ramified quadratic extension ofK; and (2) the unique quaternion division algebra overK

Alien Registration- Robinson, Margaret M. (Portland, Cumberland County)

https://digitalmaine.com/alien_docs/24271/thumbnail.jp

Counting Fixed Points, Two-Cycles, and Collisions of the Discrete Exponential Function using p-adic Methods

Brizolis asked for which primes p greater than 3 does there exist a pair (g, h) such that h is a fixed point of the discrete exponential map with base g, or equivalently h is a fixed point of the discrete logarithm with base g. Zhang (1995) and Cobeli and Zaharescu (1999) answered with a "yes" for sufficiently large primes and gave estimates for the number of such pairs when g and h are primitive roots modulo p. In 2000, Campbell showed that the answer to Brizolis was "yes" for all primes. The first author has extended this question to questions about counting fixed points, two-cycles, and collisions of the discrete exponential map. In this paper, we use p-adic methods, primarily Hensel's lemma and p-adic interpolation, to count fixed points, two cycles, collisions, and solutions to related equations modulo powers of a prime p.Comment: 14 pages, no figure

Rethinking the Composition of a Rational Antibiotic Arsenal for the 21st Century

The importance of the human microbiome in health may be the single most valuable development in our conception of the microbial world since Pasteur\u27s germ theory of the 1860s. Its implications for our understanding of health and pathogenesis are profound. Coupled with the revolution in diagnostics that we are now witnessing - a revolution that changes medicine from a science of symptoms to a science of causes - we cannot continue to develop antibiotics as we have for the past 80 years. Instead, we need to usher in a new conception of the role of antibiotics in treatment: away from single molecules that target broad phylogenetic spectra and towards targeted molecules that cripple the pathogen while leaving the rest of the microbiome largely intact

The H-alpha and Infrared Star Formation Rates for the Nearby Field Galaxy Survey

We investigate the H-alpha and infrared star formation rate (SFR) diagnostics for galaxies in the Nearby Field Galaxy Survey (NFGS). For the 81 galaxies in our sample, we derive H-alpha fluxes (included here) from integrated spectra. There is a strong correlation between the ratio of far-infrared to optical luminosities L(FIR)/L(H-alpha) and the extinction E(B-V) measured with the Balmer decrement. Before reddening correction, the SFR(IR) and SFR(H-alpha) are related to each other by a power-law. Correction of the SFR(H-alpha) for extinction using the Balmer decrement and a classical reddening curve both reduces the scatter in the SFR(IR)-SFR(H-alpha) correlation and results in a much closer agreement (within ~10%) between the two SFR indicators. This SFR relationship spans 4 orders of magnitude and holds for all Hubble types with IRAS detections in the NFGS. A constant ratio between the SFR(IR) and SFR(H-alpha) for all Hubble types, including early types (S0-Sab), suggests that the IR emission in all of these objects results from a young stellar population.Comment: 23 pages, 5 figures, 1 table. Accepted for publication in the Astronomical Journal. V2: Important changes: IRAS fluxes updated. Only moderate and good quality IRAS FIR fluxes are now used, resulting in slight changes to the equations and figures. The IR and H-alpha SFRs now agree to within ~10%, rather than ~30% as quoted previousl

The actin networks of chytrid fungi reveal evolutionary loss of cytoskeletal complexity in the fungal kingdowm

Cells from across the eukaryotic tree use actin polymer networks for a wide variety of functions, including endocytosis, cytokinesis, and cell migration. Despite this functional conservation, the actin cytoskeleton has undergone significant diversification, highlighted by the differences in the actin networks of mammalian cells and yeast. Chytrid fungi diverged before the emergence of the Dikarya (multicellular fungi and yeast) and therefore provide a unique opportunity to study actin cytoskeletal evolution. Chytrids have two life stages: zoospore cells that can swim with a flagellum and sessile sporangial cells that, like multicellular fungi, are encased in a chitinous cell wall. Here, we show that zoospores of the amphibian-killing chytrid Batrachochytrium dendrobatidis (Bd) build dynamic actin structures resembling those of animal cells, including an actin cortex, pseudopods, and filopodia-like spikes. In contrast, Bd sporangia assemble perinuclear actin shells and actin patches similar to those of yeast. The use of specific small-molecule inhibitors indicate that nearly all of Bd’s actin structures are dynamic and use distinct nucleators: although pseudopods and actin patches are Arp2/3 dependent, the actin cortex appears formin dependent and actin spikes require both nucleators. Our analysis of multiple chytrid genomes reveals actin regulators and myosin motors found in animals, but not dikaryotic fungi, as well as fungal-specific components. The presence of animal- and yeast-like actin cytoskeletal components in the genome combined with the intermediate actin phenotypes in Bd suggests that the simplicity of the yeast cytoskeleton may be due to evolutionary loss

Flight and Integrated Testing: Blazing the Trail for the Ares Launch Vehicles

It has been 30 years since the United States last designed and built a human-rated launch vehicle. The National Aeronautics and Space Administration (NASA) has marshaled unique resources from the government and private sectors that will carry the next generation of astronauts into space safer and more efficiently than ever and send them to the Moon to develop a permanent outpost. NASA's Flight and Integrated Test Office (FITO) located at Marshall Space Flight Center and the Ares I-X Mission Management Office have primary responsibility for developing and conducting critical ground and flight tests for the Ares I and Ares V launch vehicles. These tests will draw upon Saturn and the Space Shuttle experiences, which taught the value of using sound systems engineering practices, while also applying aerospace best practices such as "test as you fly" and other lessons learned. FITO will use a variety of methods to reduce the technical, schedule, and cost risks of flying humans safely aboard a launch vehicle