Introduction to logical entropy and its relationship to Shannon entropy

We live in the information age. Claude Shannon, as the father of the information age, gave us a theory of communications that quantified an “amount of information,” but, as he pointed out, “no concept of information itself was defined.” Logical entropy provides that definition. Logical entropy is the natural measure of the notion of information based on distinctions, differences, distinguishability, and diversity. It is the (normalized) quantitative measure of the distinctions of a partition on a set-just as the Boole–Laplace logical probability is the normalized quantitative measure of the elements of a subset of a set. And partitions and subsets are mathematically dual concepts – so the logic of partitions is dual in that sense to the usual Boolean logic of subsets, and hence the name “logical entropy.” The logical entropy of a partition has a simple interpretation as the probability that a distinction or dit (elements in different blocks) is obtained in two independent draws from the underlying set. The Shannon entropy is shown to also be based on this notion of information-as-distinctions; it is the average minimum number of binary partitions (bits) that need to be joined to make all the same distinctions of the given partition. Hence all the concepts of simple, joint, conditional, and mutual logical entropy can be transformed into the corresponding concepts of Shannon entropy by a uniform non-linear dit-bit transform. And finally logical entropy linearizes naturally to the corresponding quantum concept. The quantum logical entropy of an observable applied to a state is the probability that two different eigenvalues are obtained in two independent projective measurements of that observable on that state.

Safety and Efficacy of amplitude-modulated radiofrequency electromagnetic fields in advanced hepatocellular carcinoma

Importance: Hepatocellular carcinoma (HCC) is the third leading cause of cancer death worldwide. Despite the recent approval of several new agents, long-term disease control remains elusive for most patients. Administration of 27.12 MHz radiofrequency (RF) electromagnetic fields (EMF) by means of a spoon-shaped antenna (TheraBionic P1 device) placed on the anterior part of the tongue results in systemic delivery of low and safe levels of RF EMF from head to toe. Objective: To report treatment outcomes and adverse events associated with treatment with the TheraBionic P1 device in comparison to suitable historical placebo and actively treated controls. Design: Pooled case series with comparison to historical controls. Participants: Patients with advanced HCC receiving this treatment, 18 real-world patients and 41 patients from a previously reported phase II study. Historical controls from previously conducted clinical trials. Interventions: Three hours daily treatment with the TheraBionic P1 device compared with standard of care as received by historical controls in the previously conducted trials. Main outcomes and measures: Overall survival (OS), time to progression, response rate, and adverse events in the combined pooled patients and in appropriate subgroups comparable to the historical control groups. Results: In the pooled treatment group, median OS of patients with Child-Pugh A disease (n = 32) was 10.36 (95% CI 5.42–14.07) months, 4.44 (95% CI 1.64–7.13) months for patients with Child-Pugh B disease (n = 25), and 1.99 (95% CI 0.76–3.22) months for patients with Child-Pugh C disease (n = 2). Median OS for Child-Pugh A patients was 2.62 (33.9%) months longer than the 7.74 months OS of comparable historical controls (p = 0.036). The 4.73 (95% CI 1.18–8.28) months median OS for Child-Pugh B patients receiving TheraBionic P1 device as first line therapy is slightly higher than the 4.6 months median OS of historical controls receiving Sorafenib as first line therapy. Only grade 1 mucositis and fatigue were reported by patients using the device, even among Child-Pugh B and C patients. No patients discontinued treatment because of adverse events. Conclusions and Relevance: Treatment of advanced HCC with the TheraBionic P1 device is well tolerated, even in patients with severely impaired liver function, and results in improved overall survival compared to historical controls without any significant adverse events, even after many years of continuous treatment. This treatment modality appears to be well suited for patients who have failed or are intolerant to currently approved therapies.

Recent developments in mitochondrial medicine (Part 1)

Research into elucidating structure and function of mitochondria has been quite steady between the time of discovery during the end of the 19th century until towards the late 1980’s. During the 1990s there was talk about a “comeback” of this organelle reflecting a widely revitalized interest into mitochondrial research which was based on two major discoveries made during that time. The first was the etiological association between human diseases and mitochondrial DNA mutations, while the second revealed the crucial function of mitochondria during apoptosis. The March 5th, 1999 issue of Science even featured a textbook image of a mitochondrion on its front cover and was entirely dedicated to this organelle. Whilst the term “comeback” might have been appropriate to describe the general excitement surrounding the new mitochondrial discoveries made during the 1990s, a term for describing the progress made in mitochondrial research during the last two decades is difficult to find. Between 2000 and 2020 the number of publications on mitochondria has skyrocketed. It is now widely accepted that there hardly exists any human disease for which either the etiology or pathogenesis does not seem to be associated with mitochondrial malfunction. In this review we will discuss and follow several lines of mitochondrial research from their early beginnings up to the present. We hope to be able to convince the reader of what we expressed about a decade ago, that the future of medicine will come through mitochondria.

Fundamental in vitro 3D human skin equivalent tool development for assessing biological safety and biocompatibility – towards alternative for animal experiments

Nowadays, human skin constructs (HSCs) are required for biomaterials, pharmaceuticals and cosmetics in vitro testing and for the development of complex skin wound therapeutics. In vitro three-dimensional (3D) dermal-epidermal based interfollicular, full-thickness, human skin equivalent (HSE) was here developed, recapitulating skin morphogenesis, epidermal differentiation, ultra-structure, tissue architecture, and barrier function properties of human skin. Different 3D cell culture conditions were tested to optimize HSE maturation, using various commercially available serum/animal component-free and/or fully defined media, and air-liquid interface (ALI) culture. Optimized culture conditions allowed the production of HSE by culturing normal human dermal fibroblasts (NHDFs) for 5–7 days in CELLnTEC-Prime Fibroblast (CnT-PR-F) medium and then culturing normal human epidermal keratinocytes (NHEKs) for 3 days in CELLnTEC-Prime Epithelial culture (CnT-PR) medium on them. Co-culture was then submerged overnight in CELLnTEC-Prime-3D barrier (CnT-PR-3D) medium to stimulate cell-cell contact formation and finally placed at ALI for 15–20 days using CnT-PR-3D medium. Histological analysis revealed uniform distribution of NHDFs in the dermal layer and their typical elongated morphology with filopodia. Epidermal compartment showed a multi-layered structure, consisting of stratum basale, spinosum, granulosum, and corneum. NHDFs and keratinocytes of basal layer were positive for the proliferation marker Kiel 67 (Ki-67) demonstrating their active state of proliferation. The presence of typical epidermal tissue proteins (keratins, laminins, filaggrin, loricin, involucrin, and β-tubulin) at their correct anatomical position was verified by immunohistochemistry (IHC). Moreover, transmission electron microscopy (TEM) analyses revealed basement membrane with lamina lucida, lamina densa, hemidesmosomes and anchoring fibers. The epidermal layers showed abundant intracellular keratin filaments, desmosomes, and tight junction between keratinocytes. Scanning electron microscopy (SEM) analyses showed the interwoven network of collagen fibers with embedded NHDFs and adjacent stratified epidermis up to the stratum corneum similar to native human skin. HSE physiological static contact angle confirmed the barrier function. The developed HSE represents a fundamental in vitro tool to assess biocompatibility of biomaterials, pharmacotoxicity, safety and effectiveness of cosmetics, as well as to investigate skin biology, skin disease pathogenesis, wound healing, and skin infection.

COVID-19: SARS-CoV-2 susceptibility in healthcare workers – cluster study at a German Teaching Hospital

Purpose: The local health department (in German: Gesundheitsamt) ordered a shutdown of a teaching hospital due to the severe acute respiratory syndrome coronavirus (SARS-CoV-2) outbreak – one index patient and five infected healthcare workers – and put it under quarantine. For the first time, all patients plus all employees of one German hospital (healthcare providers, physicians, and nurses) were tested to detect silent or asymptomatic carriers. Methods: A serial polymerase chain reaction (PCR) test for SARS-CoV-2 was performed three times (1) between April 3rd and 5th, 2020 [n = 1171], (2) between April 8th and 9th, 2020 [n = 953] and (3) between April 14th and 17th, 2020 [n = 983]. Results: The teaching hospital’s proven coronavirus disease 2019 (COVID-19) patient load on Friday, April 3rd, 2020, was 34 patients, of whom 11 were on ventilation in the intensive care unit (ICU), one in the intermediate care unit (IMC), and 22 in the infectious disease ward. Another 32 patients in a different infectious disease ward were suspected for COVID-19 with test results pending. COVID-19 silent carrier (asymptomatic) positivity rates based on the phases of testing were (1) n = 24 (2.1%), (2) n = 25 (2.6%) and n = 9 (0.9%). The cumulative infection rate for healthcare providers, physical therapists, physicians, and nurses was 1.8%, 4.5%, 4.8%, and 11.9% which were associated with the type and extent of COVID-19 patient contact (p < 0.05). Conclusion: Despite prior proper preparation, a COVID-19 positive patient load of up to 34.8% (46 of 132 hospital beds) resulted in a 10- to 20-fold increase in risk for healthcare workers for SARS-CoV-2 compared to the general population. Because of asymptomatic carriers, a COVID-19-free hospital cannot be expected to exist. Based on our experience, repeated testing of all staff members with patient contact is necessary and is the best option to effectively contain the virus. Those having the most contact with patients had the highest risk of becoming infected (10- to 20-fold higher risk), with nurses being at the highest risk.

Multi-messengers at ultra-high energies with the Pierre Auger Observatory

The Pierre Auger Observatory is an Ultra-High Energy Cosmic Ray (UHECR) detector which has studied cosmic particles with energies above and around 1018 eV for more than 15 years. It has proved to be the most competitive instrument at these energies and has produced a wealth of valuable results, improving our understanding of UHECRs. A complete understanding of these highest energy particles is crucial to understand the extreme astrophysical events in which they are produced and accelerated, as well as their propagation to Earth. In the same range of energies, UHE photons and neutrinos are of paramount importance as, being electrically neutral, they point back to their origin while charged particles are deflected in the galactic and extragalactic magnetic fields. The flux of extragalactic photons, neutrinos, and cosmic rays are believed to be highly linked, by their origin and their interactions. Each messenger provides different information about the potential sources, and having detection means for all four messengers, including gravitational waves, allows us to shed light on energetic sources of astroparticles. The Pierre Auger Observatory benefits from a large exposure and a good angular resolution, and is efficient in detecting UHE photons and neutrinos. These performances make possible follow-up searches for events detected by gravitational waves, such as the binary mergers observed by the LIGO/Virgo detectors, or any other energetic sources of particles.

Partition Markov Model for Covid-19 Virus

In this paper, we investigate a specific structure within the theoretical framework of Partition Markov Models (PMM) [see García Jesús and González-López, Entropy 19, 160 (2017)]. The structure of interest lies in the formulation of the underlying partition, which defines the process, in which, in addition to a finite memory o associated with the process, a parameter G is introduced, allowing an extra dependence on the past complementing the dependence given by the usual memory o. We show, by simulations, how algorithms designed for the classic version of the PMM can have difficulties in recovering the structure investigated here. This specific structure is efficient for modeling a complete genome sequence, coming from the newly decoded Coronavirus Covid-19 in humans [see Wu et al., Nature 579, 265–269 (2020)]. The sequence profile is represented by 13 units (parts of the state space’s partition), for each of the 13 units, their respective transition probabilities are computed for any element of the genetic alphabet. Also, the structure proposed here allows us to develop a comparison study with other genomic sequences of Coronavirus, collected in the last 25 years, through which we conclude that Covid-19 is shown next to SARS-like Coronaviruses (SL-CoVs) from bats specimens in Zhoushan [see Hu et al., Emerg Microb Infect 7, 1–10 (2018)].

A survey on pseudo-Chebyshev functions

In recent articles, by using as a starting point the Grandi (Rhodonea) curves, sets of irrational functions, extending to the fractional degree the 1st, 2nd, 3rd and 4th kind Chebyshev polynomials have been introduced. Therefore, the resulting mathematical objects are called pseudo-Chebyshev functions. In this survey, the results obtained in the above articles are presented in a compact way, in order to make the topic accessible to a wider audience. Applications in the fields of weighted best approximation, roots of 2 × 2 non-singular matrices and Fourier series are derived.

A copula-based quantifying of the relationship between race inequality among neighbourhoods in São Paulo and age at death

In this paper, we combine two statistical tools with the objective of creating models that represent the dependence between (i) the proportion of the black/brown population in relation to the total population of a neighborhood (pct) and (ii) the average age at which people died in the neighborhood (age). We explore the dependence between pct and age in São Paulo city, Brazil, during 2018. The statistical tools are models of copulas and informative and non-informative settings according to the Bayesian perspective. The different scenarios and models allow us to delineate the dependence between pct and age, and, through the Bayesian Information Criterion we can indicate which of these models best represents the data. The approach implemented here allows us to define estimates of variations in life expectancy conditioned by percentage intervals of pct. With them, we can conclude that on average all the scenarios point to a decrease in life expectancy by increasing the proportion of pct. When conditioning the percentages of pct to 4 intervals (0, 0.25], (0.25, 0.5], (0.5, 0.75], (0.75, 1] respectively, we note that the expectation is reduced in average at a constant rate from one interval in comparison with the immediate and next interval from left to right in [0, 1].

A copula based representation for tailings dam failures

In this article, we model the dependence between dam factor and D max, where dam factor is an indicator of risk of a tailings dam failure, which involves the height H of the tailings dam, the volume of material housed by the tailings dam VT and the volume dispensed by the tailings dam, VF, when the dam breaks. And, Dmax is the maximum distance traveled by the material released by the tailings dam, after the collapse. With the dependence found via copula models and Bayesian estimation, given a range of dam factor, we estimate the probability of the released material to exceed a certain threshold. Since the dam factor involves the released volume VF (unknown before the dam break), we present a naive way to estimate it using VT and H. In this way, it is possible to estimate the dam factor of a tailings dam and with such a value to identify the probability of the tailings dam to show a Dmax that exceeds a certain threshold.