Much previous study has been done on the degree spectra of prime models of a complete atomic decidable theory. Here we study the analogous questions for homogeneous models. We say a countable model A has a d-basis if the types realized in A are all computable and the Turing degree d can list ∆0-indices for all types realized in A. We say A has a d-decidable copy if there exists a model B ∼= A s...

We prove in this note the following result. Let n > 1 be an integer and let R be an n!torsion-free semiprime ring with identity element. Suppose that there exists an additive mapping D : R→ R such that D(xn) =∑nj=1 xn− jD(x)x j−1is fulfilled for all x ∈ R. In this case, D is a derivation. This research is motivated by the work of Bridges and Bergen (1984). Throughout, R will represent an associ...

Define the Apéry polynomial of degree n by A n (x) = n k=0 n k 2 n + k k 2. We determine p−1 k=0 (−1) k A k (1/4) and p−1 k=0 (−1) k A k (1/16) modulo a prime p > 3. Let b and c be integers and let the generalized trinomial coefficient T n (b, c) be the coefficient of x n in the expansion of (x 2 +bx+c) n. We establish the following new congruence p−1 k=0 T k (b, c) 2 (b 2 − 4c) k ≡ c(b 2 − 4c)...

Let p ≡ 7 mod 16 be a prime. Then there are integers a, b, c, d with a ≡ 15 mod 16, b ≡ 0 mod 4, p = a + 2(b + c + d), and 2ab = c − 2cd − d. We show that there is a regular Hadamard matrix of order 4p provided that p = a± 2b or p = a+ δ1b+4δ2c+4δ1δ2d with δi = ±1. ∗This research was done during a visit of the first two authors at the University of Augsburg

Given any natural number q > 3 we show there exists an integer t ≤ [2 log2 (q – 3)] such that an Hadamard matrix exists for every order 2q where s > t. The Hadamard conjecture is that s = 2. This means that for each q there is a finite number of orders 2q for which an Hadamard matrix is not known. This is the first time such a statement could be made for arbitrary q. In particular it is already...