Atkin’s algorithm [2] for computing square roots in Z∗ p , where p is a prime such that p ≡ 5mod 8, has been extended by Müller [15] for the case p ≡ 9mod 16. In this paper we extend Atkin’s algorithm to the general case p ≡ 2+1mod 2, for any s ≥ 2, thus providing a complete solution for the case p ≡ 1 mod 4. Complexity analysis and comparisons with other methods are also provided.

We construct a p-adic version of Elliptic Cohomology whose coefficient ring agrees with Serre’s ring of p-adic modular forms. We then construct a stable operation Ûp in this theory agreeing with Atkin’s operator Up on p-adic modular forms. Throughout the paper we assume given a fixed prime p ≥ 5. We begin as in [2] by considering the universal Weierstrass cubic (for Z(p) algebras) Ell/R∗: Ell:Y...

Mango and Moringa are nutrient-dense foods. This study examined two mango varieties (apple Tommy Atkin) for nectar production fortifying with leaf extract to reduce postharvest losses, food insecurity, malnutrition. Post-harvest losses high in Kenya (40-50%) affects farmers' income the environment if processed into nectar, can address (PHL). is rich vitamins minerals but lacks adequate micronut...

We compute the standard Euclidean window of hadronic vacuum polarization using multiple independent blinded analyses. improve continuum and infinite-volume extrapolations dominant quark-connected light-quark isospin-symmetric contribution address additional subleading systematic effects from sea-charm quarks residual chiral-symmetry breaking first principles. find ${a}_{\mathrm{\ensuremath{\mu}...

509 Prognostic Significance of F-FDG Uptake in Hepatocellular Carcinoma Treated with Transarterial Chemoembolization or Concurrent Chemoradiotherapy: A Multicenter Retrospective Cohort Study Jeong Won Lee, Jin Kyoung Oh, Yong An Chung, Sae Jung Na, Seung Hyup Hyun, Il Ki Hong, Jae Seon Eo, Bong-Il Song, Tae-sung Kim, Do Young Kim, Seung Up Kim, Dae Hyuk Moon, Jong Doo Lee, and Mijin Yun 517 Saf...

Nigel T. Bishop, Roberto Gómez, Luis Lehner, Manoj Maharaj, and Jeffrey Winicour Department of Mathematical Sciences, University of South Africa, P.O. Box 392, Pretoria 0003, South Africa Pittsburgh Supercomputing Center, 4400 Fifth Avenue, Pittsburgh, Pennsylvania 15213, USA Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA Department of Physics...

W. Barreto, A. Da Silva, R. Gómez, L. Lehner, L. Rosales, and J. Winicour Centro de Fı́sica Fundamental, Facultad de Ciencias, Universidad de los Andes, Mérida, Venezuela Department of Physics and Astronomy, 2002 Nicholson Hall, Louisiana State University, Baton Rouge, Louisiana 70803-4001, USA Pittsburgh Supercomputing Center, 4400 Fifth Avenue, Pittsburgh, Pennsylvania 15213, USA Department of...

n>0 (1− q) be Dedekind’s eta-function. For a prime p, denote by U Atkin’s Up-operator. We say that a function φ with a Fourier expansion φ = ∑ u(n)q is congruent to zero modulo a power of a prime p, φ = ∑ u(n)q ≡ 0 mod p if all its Fourier expansion coefficients are divisible by this power of the prime; u(n) ≡ 0 mod p for all n. In this paper we prove the following congruences. Theorem 1. (i) I...

This paper describes the concept of a mobile Information System which will be implemented at the Faculty of Business, Economics and Information Systems at the University of Regensburg. The prototype „U-Know“ („Ubiquitous Knowledge Management“) will enable typical functions of Knowledge Management Systems for mobile devices such as Mobile Phones and Personal Digital Assistants. In this way facul...