To the great loss of the scienti)c community, Professor Helmut Thiele has passed away unexpectedly, but peacefully on Friday, January 10, 2003 in his home town of Cologne. He was born on June 20, 1926 in Saratoga. In 1946, he took up a profession as “Neulehrer”, a specially created position as school teacher with a short training, to make up for the lack of teachers after World War II. From 194...

If a mod m is a unit then aφ(m) ≡ 1 mod m by Euler’s theorem. (When m = p is prime, then this becomes ap−1 ≡ 1 mod p for a 6≡ 0 mod p, which is Fermat’s little theorem.) Depending on a, it might happen that an ≡ 1 mod m for an n that is smaller than φ(m). Example 1.1. Take m = 11. Then a10 ≡ 1 mod 11 for all a 6≡ 0 mod 11. When a = 2 no exponent smaller than 10 works, but when a = 3 we can use ...

In this paper, we prove arithmetic properties modulo 5 and 7 satisfied by the function pod(n) which denotes the number of partitions of n wherein odd parts must be distinct (and even parts are unrestricted). In particular, we prove the following: For all n ≥ 0, pod(135n + 8) ≡ 0 (mod 5), pod(135n + 107) ≡ 0 (mod 5), pod(135n + 116) ≡ 0 (mod 5), pod(675n + 647) ≡ 0 (mod 25), pod(3375n + 1997) ≡ ...

Abstract Prolonged drought periods cause a cascade of biochemical alterations in plants and lead to reduction plant growth crop productivity. Seaweed extracts are category biostimulants that effective alleviating stress on plants. However, the effect seaweed attenuating negative impact Faba bean ( Vicia faba L .) under arid semi-arid conditions has received little attention. This study was cond...

On primitive roots of 1 mod p k , divisors of p ± 1, Wieferich primes, and quadratic analysis mod p Abstract Primitive roots of 1 mod p k (k > 2 and odd prime p) are sought, in cyclic units group G k ≡ A k B k mod p k , coprime to p, of order (p − 1)p k−1. 'Core' subgroup A k has order p − 1 independent of precision k, and 'extension' subgroup B k of all p k−1 residues 1 mod p is generated by p...

Given a nite dimensional algebra over an algebraically closed eld one frequently disregards the projective objects in the category mod of nite dimensional-modules and focuses on the stable category mod. The objects of mod are those of mod but the Hom-groups are the ordinary Hom-groups modulo the subgroup of those morphisms which factor through a projective-module. In this paper we show that mod...

A 3 copies of the manuscript A please indicate the corresponding author's details on the manuscript (tel/fax/e-mail) A 3 sets of figures A electronic version of the manuscript (Label the floppy disk with the type of computer and word processing package used, your name and the name of the file on the disk.) A electronic version of the figures For Review Papers: A 3 copies of the manuscript A ple...

A 4 copies of the manuscript A please indicate the corresponding author's details on the manuscript (tel/fax/e-mail) A 4 sets of figures A electronic version of the manuscript (Label the floppy disk with the type of computer and word processing package used, your name and the name of the file on the disk.) A electronic version of the figures (optional, see instructions below) For Review Papers:...

A 3 copies of the manuscript A please indicate the corresponding author's details on the manuscript (tel/fax/e-mail) A 3 sets of figures A electronic version of the manuscript (Label the floppy disk with the type of computer and word processing package used, your name and the name of the file on the disk.) A electronic version of the figures For Review Papers: A 3 copies of the manuscript A ple...