Let (G, +) be a

groupoid. Then (RNQ(G), [symmetry]) is a group with identity element

Among the topics are the Atiyah-Singer cobordism invariance and the tangent

groupoid, multiplicative ergodic theorems for transfer operators: towards the identification and analysis of coherent structures in non-autonomous dynamical systems, two proofs of Taubes' theorem on strictly ergodic flows, extremal higher co-dimension cycles of the space of complete conics, and a general solution to (free) deterministic equivalents.

which do not have the "morphism of

groupoid" property such as

The Weyl

groupoid of [B.sub.q] is a

groupoid, denoted [W.sub.q],

A

groupoid S is called a left almost semigroup if it satisfies the following left invertive law, (ab)c = (cb)a, for all a, b, c [member of] S.

Abel Grassmann's

groupoid abbreviated as an AG-groupoid is a

groupoid whose element satisfies the left invertive law i.e (ab)c = (cb)a for all a,b,c [member of] S.

A

groupoid is a small category in which all morphisms are invertible.

A nonempty set R with an m-ary operation f is called an m-ary

groupoid and is denoted by (R, f) (see Dudek [24]).

A non-empty set of elements G is said to form a

groupoid if in G is defined a binary operation called the product denoted by * such that a * b [member of] G, [for all] a, b [member of] G.

Then C(S)(e, f) = {(f, ([u.sub.2], [v.sub.1]), e)} and C(S) is a

groupoid. It is therefore easy to see that any mapping C(E) [right arrow] C(S) is a Pos-equivalence.

Keywords: frozen accident, rate distortion function, protein folding, free energy density, spin glass,

groupoid, Onsager relations, holonomy

An Abel- Grassmann's

groupoid, abbreviated as an AG-groupoid (or in some papers left almost semigroup), is a non-associative algebraic structure mid way between a

groupoid and a commutative semigroup.