In this paper, the ambiguity of nite state irreducible Markov chain trajectories is reminded and is obtained for two state Markov chain. I give an applicable example of this concept in President election

Let $E$ be an elliptic curve over a number field $K$ defined by monic irreducible cubic polynomial $F(x)$. When is \textit{nice} at all finite primes of $K$, we bound its $2$-Selmer rank in terms the $2$-rank modified ideal class group $L=K[x]/{(F(x))}$, which call \textit{semi-narrow group} $L$. We then provide several sufficient conditions for being nice prime. As application, when real quadr...

A connected graph on three or more vertices is said to be irreducible if it has no leaves, and if each vertex has a unique neighbor set. A connected graph on one or two vertices is also said to be irreducible, and a disconnected graph is irreducible if each of its connected components is irreducible. In this paper, we study the class of irreducible graphs. In particular, we consider an algorith...

Consider an ideal I ⊂ R = C[x] = C[x 1 ,. .. , x n ] defining a complex affine variety X = V (I) = {x ∈ C n | ∀f ∈ I, f (x) = 0}. We describe the associated components VAss(I) = {V (P) | P ∈ Ass(I)} by means of numerical primary decomposition (NPD). The method is based on the construction of deflation ideal I (d) that defines the deflated variety X (d) = V (I (d)) in a higher-dimensional comple...

The Schubert cells eλ are the orbits of the Iwahori subgroup B̃, while the Birkhoff strata Sλ are the orbits of the opposite Iwahori subgroup B̃ −. The cells and the strata are dual in the sense that Sλ ∩ eλ = {λ}, and the intersection is transverse. The closure of eλ is the affine Schubert variety Xλ. It has dimension ` (λ), where ` is the minimal length occuring in the coset λW̃I , and its cells...

in this paper we study the notions of cogenerator and subdirectlyirreducible in the category of s-poset. first we give somenecessary and sufficient conditions for a cogenerator $s$-posets.then we see that under some conditions, regular injectivityimplies generator and cogenerator. recalling birkhoff'srepresentation theorem for algebra, we study subdirectlyirreducible s-posets and give this theo...