0. Introduction 2 1. Stability and ranks 3 PART I: INDEPENDENCE 8 2. Forking 8 3. Indiscernible sets 9 4. Finite equivalence relations 12 5. Further properties of forking 16 6. An example of the use of forking 19 PART II: PRIME MODELS 22 7. General isolation notion 22 8. Examples of isolation notions 26 9. Spectrum of stability 27 10. a -prime models 28 11. Structure of a -saturated models 30 1...

A bstract Employing the QCD factorization formalism we compute $$ {B}_u^{-}\to {\gamma}^{\ast}\mathrm{\ell}{\overline{v}}_{\mathrm{\ell}} B u − → γ ∗ ℓ <mml:...

n = 1: 1 = 0 + 1; n = 2 (prime): 2 = 1 + 1; n = 3 (prime) is not a sum of two squares. n = 4: 4 = 2 + 0. n = 5 (prime): 5 = 2 + 1. n = 6 is not a sum of two squares. n = 7 (prime) is not a sum of two squares. n = 8: 8 = 2 + 2. n = 9: 9 = 3 + 0. n = 10: 10 = 3 + 1. n = 11 (prime) is not a sum of two squares. n = 12 is not a sum of two squares. n = 13 (prime): 13 = 3 + 2. n = 14 is not a sum of t...

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...

Bresar in 1993 proved that each biderivation on a noncommutative prime ring is a multiple of a commutatot. A result of it is a characterization of commuting additive mappings, because each commuting additive map give rise to a biderivation. Then in 1995, he investigated biderivations, generalized biderivations and sigma-biderivations on a prime ring and generalized the results of derivations fo...

• n is prime. Then G is cyclic by Theorem 1. • n = pc for some prime p and c > 1. In this case if there is no element of order n, then all the orders must divide pc−1. We get n = pc elements x such that xp c−1 = 1, violating (*). • n = pq for co-prime p and q. In this case let H and F be two subgroups of G defined as follows: H = {a : ap = 1} and F = {b : bq = 1}. Then |H| ≤ p < n and |F | ≤ q ...

Ramanujan’s famous partition congruences modulo powers of 5, 7, and 11 imply that certain sequences of partition generating functions tend `-adically to 0. Although these congruences have inspired research in many directions, little is known about the `-adic behavior of these sequences for primes ` ≥ 13. We show that these sequences are governed by “fractal” behavior. Modulo any power of a prim...

Let p be prime, K a field of characteristic 0. Let (x1, . . . , xn) ∈ Kn such that xi 6= 0 for all i and xi/xj is not a root of unity for all i 6= j. We prove that there exist integers 0 < e1 < e2 < · · · < en such that det (x ei j ) 6= 0. The proof uses p-adic arguments. As corollaries, we derive the linear independence of certain Witt vectors and study the result of applying the Witt functor ...

We have already learned ideal of a ring. A lattice is an algebraic structure which looks like a ring for both have two operations. Similarly, a lattice also has a special structure named as the same as ideal. Definition 1. A subset I of a lattice L is an ideal if it is a sublattice of L and x ∈ I and a ∈ L imply that x ∩ a ∈ I. Specially, A proper ideal I of L is prime if a, b ∈ L and a ∩ b ∈ I...