In this paper, we obtain the Rényi entropy rate for irreducible-aperiodic Markov chains with countable state space, using the theory of countable nonnegative matrices. We also obtain the bound for the rate of Rényi entropy of an irreducible Markov chain. Finally, we show that the bound for the Rényi entropy rate is the Shannon entropy rate.

Let X be a Hausdorff quotient of standard space (that is locally compact separable metric space). It shown that the following are equivalent: (i) image an irreducible map from space; (ii) has sequentially dense subset satisfying two technical conditions involving double sequences; (iii) whenever q : Y → , restriction ⁎ | V onto (where pure derived and closure set singleton fibres ). The proof u...

in this paper we introduce two-wavelet constants for square integrable representations of homogeneous spaces. we establishthe orthogonality relations for square integrable representationsof homogeneous spaces which give rise to the existence of aunique self adjoint positive operator on the set of admissiblewavelets. finally, we show that this operator is a constant multiple of identity operator...

Let G be a complex reductive algebraic group and let Γ be a finitely generated group. In this paper we study properties of irreducible and completely reducible representations ρ : Γ → G in the context of the geometric invariant theory of the G-action on Hom(Γ, G) by conjugation. In particular, we prove that ρ is poly-stable (i.e. its orbit is closed) if and only if ρ is completely reducible. We...

In this article, we introduce monomial irreducible representations of the special linear Lie algebra $sln$. We will show that this kind of representations have bases for which the action of the Chevalley generators of the Lie algebra on the basis elements can be given by a simple formula.

Abstract Let $${\mathcal {C}} \subset {\mathbb {N}}^p$$ C ⊂ N p be a finitely generated integer cone and $$S\subset {\mathcal {C}}$$ S an affine semigroup such that the real cones by S are equal. ...

We study the conormal sheaves and singular schemes of 1-dimensional foliations on smooth projective varieties $X$ dimension 3 Picard rank 1. prove that if scheme has 0, then sheaf is $\mu$-stable whenever tangent bundle $TX$ stable, apply this fact to characterization certain irreducible components moduli space 2 reflexive $\mathbb{P}^3$ a quadric hypersurface $Q_3\subset\mathbb{P}^4$. Finally,...

For a general cubic fourfold $X\subset\mathbb{P}^5$ with Fano scheme of lines $F$, we prove number properties the universal family $I\to F$ and various subloci. We first describe moduli ramification theory genus four fibration $p:I\to X$ explore its relation to birational model $F$ in $I$. The main part paper is devoted describing locus $V\subset triple lines, i.e., fixed Voisin map $\phi:F\das...

We examine the structure of the Weierstrass semigroup of an m-tuple of points on a smooth, projective, absolutely irreducible curve X over a finite field IF. A criteria is given for determining a minimal subset of semigroup elements which generate such a semigroup where 2 ≤ m ≤| IF |. For all 2 ≤ m ≤ q + 1, we determine the Weierstrass semigroup of any m-tuple of collinear IFq2 -rational points...