On the supercongruence conjectures of van Hamme

In 1997, van Hamme developed $p-$adic analogs, for primes $p$, of several series which relate hypergeometric series to values of the gamma function, originally studied by Ramanujan. These analogs relate truncated sums of hypergeometric series to values of the $p-$adic gamma function, and are called Ramanujan type supercongruences. In all, van Hamme conjectured 13 such formulas, three of which were proved by van Hamme himself, and five others have been proved recently using a wide range of methods. Here, we explore four of the remaining five van Hamme supercongruences, revisit some of the proved ones, and provide some extensions

The ALI Principles: A Farewell to Fault—But What Remedy for the Egregious Marital Misconduct of an Abusive Spouse?

The fundamental premise of this commentary is that the ALl has erred in not including appropriate nonfinancial fault-based factors in the Principles for three major reasons: 1) other no-fault laws, including no-fault automobile insurance law, no-fault workers compensation law, and strict liability in tort law, have all incorporated a number of fault-based exceptions to their general no-fault framework for serious or egregious conduct, and American divorce law should likewise have a similar fault-based exception for serious or egregious marital misconduct; 2) a substantial number of states continue to recognize and utilize a number of fault-based statutory factors in divorce for serious and egregious marital misconduct, and these state courts generally have applied such fault-based remedies in a realistic and responsible manner; and 3) alternative tort or criminal law remedies for serious and egregious marital misconduct have proven to be inadequate legal remedies in theory and practice. Accordingly, this commentary will conclude that fault-based factors for egregious marital misconduct should be retained, or should be seriously reconsidered, by any state legislature that is considering legislative adoption of the ALI Principles

Solubility of iron, nickel, and cobalt in liquid potassium and effect of oxygen gettering agents on iron solubility

Iron, nickel, and cobalt solubility in liquid potassium, and effect of oxygen gettering agents on iron solubilit

An Analysis of the Price Formation Process at a HUD Auction

This study considers whether auctioned properties sell for different prices than they would bring through private negotiation. After reviewing the procedural aspects of HUD auctions, we compare the observed prices of properties sold at one such auction with predicted market values based on assessment ratios for the region to detect any discount or premium. We also consider whether the order of sale of the auctioned properties affects observed prices. We find that sample properties sell at a significant discount relative to predicted market values and that prices tend to increase as the auction proceeds, holding quality of the properties constant.

Zeros of some level 2 Eisenstein series

The zeros of classical Eisenstein series satisfy many intriguing properties. Work of F. Rankin and Swinnerton-Dyer pinpoints their location to a certain arc of the fundamental domain, and recent work by Nozaki explores their interlacing property. In this paper we extend these distribution properties to a particular family of Eisenstein series on Gamma(2) because of its elegant connection to a classical Jacobi elliptic function cn(u) which satisfies a differential equation. As part of this study we recursively define a sequence of polynomials from the differential equation mentioned above that allow us to calculate zeros of these Eisenstein series. We end with a result linking the zeros of these Eisenstein series to an L-series.Comment: 14 pages, 1 figur

On the energy fluxes of low-energy protons and positive ions in the earth's inner radiation zones

Energy fluxes of low energy protons and positive ions in earth inner radiation zon