On the dimension of the space of integrals on coalgebras

We study the injective envelopes of the simple right $C$-comodules, and their duals, where $C$ is a coalgebra. This is used to give a short proof and to extend a result of Iovanov on the dimension of the space of integrals on coalgebras. We show that if $C$ is right co-Frobenius, then the dimension of the space of left $M$-integrals on $C$ is $\leq {\rm dim}M$ for any left $C$-comodule $M$ of finite support, and the dimension of the space of right $N$-integrals on $C$ is $\geq {\rm dim}N$ for any right $C$-comodule $N$ of finite support. If $C$ is a coalgebra, it is discussed how far is the dual algebra $C^*$ from being semiperfect. Some examples of integrals are computed for incidence coalgebras

Constructing Pointed Hopf Algebras by Ore Extensions

AbstractWe present a general construction producing pointed co-Frobenius Hopf algebras and give some classification results for the examples obtained

Virtual Addictions, Teleworking and Artificial Intelligence in the Pandemic

This article points out that during the pandemic of SARS-CoV-2 virus and quarantine, people who were already addicted to the Internet, video games or TV, this was exacerbated in most cases. Other people have had to adapt to the economic and social life generated by the pandemic, launching various online businesses. Some people, however, have developed at least one virtual addiction that has affected their health, family life, self-image, behavior, will, and psychological immunity. The article mentions the types of virtual addictions, the concept of digital dementia (Khoja, 2021) and the profile of digital addicts. Here are some tips to help you keep your balance when it comes to surfing the web. The 20-item test developed by Dr. Kimberly Young, which identifies the level of internet addiction, is mentioned. The article continues with the benefits and risks of telecommuting. The SWOT Analysis is presented as a tool for assessing strengths, weaknesses, opportunities and threats. Artificial intelligence has allocated a special space where arguments are presented that were the basis for the implementation of robot technology and digitization. Starting with the cartoon film, Wall-e, from the brief presentation of the Robot Sophia, we also argued with the help of the myth Pygmalion and Galateeia, the prudent advantages we must have in our relationship with robots. The study, conducted by Oracle and Future Workplace, highlights employees' perceptions of robots and artificial intelligence.</p

A matroid-friendly basis for the quasisymmetric functions

A new Z-basis for the space of quasisymmetric functions (QSym, for short) is presented. It is shown to have nonnegative structure constants, and several interesting properties relative to the space of quasisymmetric functions associated to matroids by the Hopf algebra morphism (F) of Billera, Jia, and Reiner. In particular, for loopless matroids, this basis reflects the grading by matroid rank, as well as by the size of the ground set. It is shown that the morphism F is injective on the set of rank two matroids, and that decomposability of the quasisymmetric function of a rank two matroid mirrors the decomposability of its base polytope. An affirmative answer is given to the Hilbert basis question raised by Billera, Jia, and Reiner.Comment: 25 pages; exposition tightened, typos corrected; to appear in the Journal of Combinatorial Theory, Series

Twisted Frobenius-Schur indicators for Hopf algebras

The classical Frobenius-Schur indicators for finite groups are character sums defined for any representation and any integer m greater or equal to 2. In the familiar case m=2, the Frobenius-Schur indicator partitions the irreducible representations over the complex numbers into real, complex, and quaternionic representations. In recent years, several generalizations of these invariants have been introduced. Bump and Ginzburg, building on earlier work of Mackey, have defined versions of these indicators which are twisted by an automorphism of the group. In another direction, Linchenko and Montgomery have defined Frobenius-Schur indicators for semisimple Hopf algebras. In this paper, the authors construct twisted Frobenius-Schur indicators for semisimple Hopf algebras; these include all of the above indicators as special cases and have similar properties.Comment: 12 pages. Minor revision

A New One-Dimensional Chaotic Map and Its Use in a Novel Real-Time Image Encryption Scheme

We present a new one-dimensional chaotic map, suitable for real-time image encryption. Its theoretical analysis, performed using some specific tools from the chaos theory, shows that the proposed map has a chaotic regime and proves its ergodicity, for a large space of values of the control parameter. In addition, to argue for the good cryptographic properties of the proposed map, we have tested the randomness of the values generated by its orbit using NIST statistical suite. Moreover, we present a new image encryption scheme with a classic bimodular architecture, in which the confusion and the diffusion are assured by means of two maps of the previously proposed type. The very good cryptographic performances of the proposed scheme are proved by an extensive analysis, which was performed regarding the latest methodology in this field