On noncommutative equivariant bundles

We discuss a possible noncommutative generalization of the notion of an equivariant vector bundle. Let $A$ be a $\mathbb{K}$-algebra, $M$ a left $A$-module, $H$ a Hopf $\mathbb{K}$-algebra, $\delta:A\to H\otimes A:=H\otimes_{\mathbb{K}} A$ an algebra coaction, and let $(H\otimes A)_\delta$ denote $H\otimes A$ with the right $A$-module structure induced by~$\delta$. The usual definitions of an equivariant vector bundle naturally lead, in the context of $\mathbb{K}$-algebras, to an $(H\otimes A)$-module homomorphism $$\Theta:H\otimes M\to (H\otimes A)_\delta\otimes_AM$$ that fulfills some appropriate conditions. On the other hand, sometimes an $(A,H)$-Hopf module is considered instead, for the same purpose. When $\Theta$ is invertible, as is always the case when $H$ is commutative, the two descriptions are equivalent. We point out that the two notions differ in general, by giving an example of a noncommutative Hopf algebra $H$ for which there exists such a $\Theta$ that is not invertible and a left-right $(A,H)$-Hopf module whose corresponding homomorphism $M\otimes H\to (A\otimes H)_\delta\otimes_AM$ is not an isomorphism.Comment: In this version we dismiss the term neb-homomorphism (hinting at 'noncommutative equivariant bundles'), as the class of modules is larger than the class of algebraic counterparts of vector bundles. We also corrected some mistakes. Our main example does not immediately extended to the left-right case and the example about the 'exotic' Hopf module works only in the left-right cas

Conservation of geometric structures for non-homogeneous inviscid incompressible fluids

We obtain a result about propagation of geometric properties for solutions of the non-homogeneous incompressible Euler system in any dimension $N\geq2$. In particular, we investigate conservation of striated and conormal regularity, which is a natural way of generalizing the 2-D structure of vortex patches. The results we get are only local in time, even in the dimension N=2; however, we provide an explicit lower bound for the lifespan of the solution. In the case of physical dimension N=2 or 3, we investigate also propagation of H\"older regularity in the interior of a bounded domain

Exact and approximate solutions for the quantum minimum-Kullback-entropy estimation problem

The minimum Kullback entropy principle (mKE) is a useful tool to estimate quantum states and operations from incomplete data and prior information. In general, the solution of a mKE problem is analytically challenging and an approximate solution has been proposed and employed in different context. Recently, the form and a way to compute the exact solution for finite dimensional systems has been found, and a question naturally arises on whether the approximate solution could be an effective substitute for the exact solution, and in which regimes this substitution can be performed. Here, we provide a systematic comparison between the exact and the approximate mKE solutions for a qubit system when average data from a single observable are available. We address both mKE estimation of states and weak Hamiltonians, and compare the two solutions in terms of state fidelity and operator distance. We find that the approximate solution is generally close to the exact one unless the initial state is near an eigenstate of the measured observable. Our results provide a rigorous justification for the use of the approximate solution whenever the above condition does not occur, and extend its range of application beyond those situations satisfying the assumptions used for its derivation.Comment: 9 pages, 6 figure

On the asymptotics of wright functions of the second kind

The asymptotic expansions of the Wright functions of the second kind, introduced by Mainardi [see Appendix F of his book Fractional Calculus and Waves in Linear Viscoelasticity (2010)], Fσ(x)=◀∑▶,Mσ(x)=∞∑n=0(−x)n/n!Γ(−nσ+1−σ)  (0&lt;σ&lt;1)   for x → ± ∞ are presented. The situation corresponding to the limit σ → 1− is considered, where Mσ(x) approaches the Dirac delta function δ(x − 1). Numerical results are given to demonstrate the accuracy of the expansions derived in the paper, together with graphical illustrations that reveal the transition to a Dirac delta function as σ → 1−.</p

Wright functions of the second kind and Whittaker functions

In the framework of higher transcendental functions, the Wright functions of the second kind have increased their relevance resulting from their applications in probability theory and, in particular, in fractional diffusion processes. Here, these functions are compared with the well-known Whittaker functions in some special cases of fractional order. In addition, we point out two erroneous representations in the literature