Let $${\mathbb {F}}_{q^n}$$ be a finite field with $$q^n$$ elements and r positive divisor of $$q^n-1$$ . An element $$\alpha \in {\mathbb {F}}_{q^n}^*$$ is called r-primitive if its multiplicative order $$(q^n-1)/r$$ Also, k-normal over {F}}_q$$ the greatest common polynomials $$g_{\alpha }(x) = \alpha x^{n-1}+ ^q x^{n-2} + \ldots ^{q^{n-2}}x ^{q^{n-1}}$$ $$x^n-1$$ in {F}}_{q^n}[x]$$ has degre...

Let R R be an algebra essentially of finite type over a field alttext="k"> k encoding="application/x-tex">k</mml:annot...

We develop a framework to analyze the dynamics of finite-dimensional quantum systemSin contact with reservoirR. The full, interactingSRdynamics is unit...

In this paper we describe the facets cone associated to transversal polymatroid presented by A = {{1, 2}, {2, 3}, . . . , {n−1, n}, {n, 1}}. Using the Danilov-Stanley theorem to characterize the canonicale module, we deduce that the base ring associated to this polymatroid is Gorenstein ring. Also, starting from this polymatroid we describe the transversal polymatroids with Gorenstein base ring...