♣ Rings . A ring is a non-empty set R with two binary operations ( + , · ) , called addition and multiplication, respectively satisfying : Axiom 1. Closure ( + ) : ∀x, y ∈ R , x + y ∈ R . Axiom 2. Commutative ( + ) : For every x, y ∈ R , x + y = y + x . Axiom 3. Associative ( + ) : ∀x, y, z ∈ R , x + (y + z) = (x + y) + z . Axiom 4. Neutral ( + ) : ∃ θ ∈ R , such that ∀x ∈ R, x + θ = θ + x = x ...

SDIV = [0..5] × Z× Z× Z× Z (where [m..n] = {m,m+1, . . . , n}) R̂DIV (pc, x, y, r, q) (pc ′, x′, y′, r′, q′) = (pc = 0) ⇒ ((pc′, x′, y′, r′, q′) = (1, x, y, x, q)) ∧ (pc = 1) ⇒ ((pc′, x′, y′, r′, q′) = (2, x, y, r, 0)) ∧ (pc = 2) ⇒ ((pc′, x′, y′, r′, q′) = if y≤r then (3, x, y, r, q) else (5, x, y, r, q)) ∧ (pc = 3) ⇒ ((pc′, x′, y′, r′, q′) = (4, x, y, (r−y), q)) ∧ (pc = 4) ⇒ ((pc′, x′, y′, r′, ...

For simplicity, we adopt the following convention: x, x0, y, y0, r are real numbers, z, z0 are elements of R2, f , f1, f2 are partial functions from R2 to R, R is a rest, and L is a linear function. Let us note that every rest is total. Let f be a partial function from R2 to R and let z be an element of R2. The functor pdiff1(f, z) yielding a function from R2 into R is defined as follows: (Def....

Let R ∈ Z be a vector with strictly positive integers entries. We denote its transpose by R = (r1, . . . , rn). In this article, unless specified otherwise, any integer vector denoted R will be assumed to have gcd(r1, . . . , rn) = 1. Let D ∈ Z. We define the degree of D as degR(D) := D ·R. When the context makes the reference to R unnecessary, we will denote degR simply by deg. The kernel of t...

SDIV = [0..5] × Z× Z× Z× Z (where [m..n] = {m,m+1, . . . , n}) R̂DIV (pc, x, y, r, q) (pc ′, x′, y′, r′, q′) = (pc = 0) ⇒ ((pc′, x′, y′, r′, q′) = (1, x, y, x, q)) ∧ (pc = 1) ⇒ ((pc′, x′, y′, r′, q′) = (2, x, y, r, 0)) ∧ (pc = 2) ⇒ ((pc′, x′, y′, r′, q′) = if y≤r then (3, x, y, r, q) else (5, x, y, r, q)) ∧ (pc = 3) ⇒ ((pc′, x′, y′, r′, q′) = (4, x, y, (r−y), q)) ∧ (pc = 4) ⇒ ((pc′, x′, y′, r′, ...

Let G be a finite group and let r be the maximum of the p-ranks of G for all primes p dividing the order to G. There exist positive integers m and n such that the exponents of Hn(G,Z) and Hm(G, Z) are greater than Suppose that G is a finite group. It has long been known that the exponent of Hn(G, Z) divides |G|, the order of G, for all n > 0. Beyond this fact very little has been established co...

It is shown that if w(z) is a finite-order meromorphic solution of the equation H(z, w)P (z,w) = Q(z,w), where P (z,w) = P (z,w(z), w(z+c1), . . . , w(z+cn)), c1, . . . , cn ∈ C, is a homogeneous difference polynomial with meromorphic coefficients, and H(z, w) = H(z, w(z)) and Q(z,w) = Q(z,w(z)) are polynomials in w(z) with meromorphic coefficients having no common factors such that max{deg w (...

The aryl hydrocarbon receptor (AHR) is a ligand-regulated transcription factor that can be activated by structurally diverse chemicals, ranging from environmental carcinogens to dietary metabolites. Evidence supporting a necessary role for the AHR in normal biology has been established; however, identification of key endogenous ligand/activator remains to be established. Here, we report the abi...