THE BINARY GOLDBACH CONJECTURE

<p>The Goldbach Conjecture, one of the oldest problems in mathematics, has fascinated and inspired many mathematicians for ages. In 1742 German mathematician Christian Goldbach, in a letter addressed to Leonhard Euler, proposed a conjecture. The modern-day version of the Binary/Strong Goldbach conjecture asserts that: Every even integer greater than 2 can be written as the sum of two primes. The conjecture had been verified empirically up to 4 × 1018, its proof however remains elusive, which seems to confirm that:</p><p><em>Some problems in mathematics remain buried deep in the catacombs of slow progress ... mind stretching mysteries await to be discovered beyond the boundaries of former thought. Avery Carr (2013) </em></p><p>The research was aimed at exposition, of the intricate structure of the fabric of the Goldbach Conjecture problem. The research methodology explores several topics, before the definite proof of the Goldbach Conjecture can be presented. The Ternary Goldbach Conjecture Corollary follows the proof of the Binary Goldbach Conjecture as well as the representation of even numbers by the difference of two primes Corollary. The research demonstrates that the Goldbach Conjecture is a genuine arithmetical question.</p>

The Binary Goldbach Conjecture

The Goldbach conjecture, one of the oldest problems in mathematics, has fascinated and inspired many mathematicians for ages. In 1742 a German mathematician Christian Goldbach, in a letter addressed to Leonhard Euler proposed a conjecture. The modern day version of the Binary/Strong Goldbach conjecture asserts that: Every even integer greater than 2 can be written as the sum of two primes. The conjecture had been empirically verified up to 4× 10 18​​​​​, its proof however remains elusive, which seems to confirm that: Some problems in mathematics remain buried deep in the catacombs of slow progress ... mind stretching mysteries await to be discovered beyond the boundaries of former thought. Avery Carr (2013) The research was aimed at exposition, of the intricate structure of the fabric of the Goldbach Conjecture problem. The research methodogy explores a number of topics, before the definite proof of the Goldbach Conjecture can be presented. The Ternary Goldbach Conjecture Corollary follows the proof of the Binary Goldbach Conjecture as well as the Representation of even numbers by the difference of two primes Corollary. The research demonstrates that the Goldbach Conjecture is a genuine arithmetical question. ​​​

Subfamily Anischiinae (Coleoptera: Eucnemidae) in Early Cretaceous of Northeast China

Rheanischia new genus, type species Rheanischia brevicornis new species (Eucnemidae, Anischiinae) is described from the Lower Cretaceous of Liaoning, China. The presence of this species in early Cretaceous deposits provides new insight into the evolution of basal lignicolous Eucnemidae clades. Both Anischiinae and Palaeoxeninae species diversified in a world dominated by gymnosperms, before the main radiation of angiosperms. More than 95% of modern eucnemid larvae have a Palaeoxenus-type highly modified head structure, but contrary to the Palaeoxenus larva, they develop in angiosperm wood. Anischiinae utilize angiosperms as well, but their head capsule shows no such modifications. These facts prove that highly specialized morphological features do not offer definite proof of similar way of life in the distant past, nor should non-modified structures be taken as proof for another kind of substrate choice. Eucnemidae have invaded angiosperms with two quite different morphological adaptations. This fact may have implications for the evolution of all clicking elateroids.

Strengthening the association of lipomatosis of nerve and nerve-territory overgrowth: a systematic review

OBJECTIVELipomatosis of nerve (LN) is a massive enlargement of a nerve due to abundant proliferation of adipose and fibrotic tissue within the epineurium—part of the spectrum of adipose lesions of nerves, including intra- and extraneural lipomas. LN has been frequently associated with soft-tissue and/or osseous overgrowth. Unfortunately, much confusion exists since many names have been used for LN (e.g., fibrolipomatous hamartoma, macrodystrophia lipomatosa, and so on). To better understand this condition and to evaluate its association with nerve-territory overgrowth, the authors attempted to compile the world’s literature on published LN cases.METHODSPubMed and Google Scholar databases were searched to identify published articles on LN cases, using a variety of terms. Publications in all languages were assessed. All publications with cases determined likely to be LN were read. Cases that provided clear clinicopathological and/or radiological evidence of LN were labeled as “definite” and cases that demonstrated features of LN (e.g., nerve-territory overgrowth) but lacked definite proof of nerve involvement were labeled as “probable.”RESULTSInitial screening revealed a total of 2465 papers. After exclusions, 281 publications reported cases with a definite diagnosis of LN and 120 articles reported cases with a probable diagnosis of LN. The authors identified 618 definite and 407 probable cases of LN. Sex distribution was balanced (51% female). Early diagnosis was common, with two-thirds of patients having symptoms in the 1st decade of life. The most commonly affected nerve was the median nerve (n = 391). Nerve-territory overgrowth was common (62% definite LN; 78% combined cases); overgrowth was exclusive to the territory of the affected nerve in all cases but 5.CONCLUSIONSThe authors present a comprehensive review and analysis of the literature of LN cases. One of the main findings was the nerve-territory overgrowth was associated with LN, especially when present earlier in life. The authors believe that all cases of LN associated with overgrowth can be explained on anatomical grounds, even in the few reported cases in which this is not immediately obvious.

Inhibition of spinal bone formation in AS: 10 years after comparing adalimumab to OASIS

A decade has passed since the publication on the comparison of the effect of adalimumab with data from a historic cohort on the progression of structural damage in the spine of patients with ankylosing spondylitis (AS). No effect could be observed, and currently, there is still no definite proof that TNF inhibitors (TNFi) inhibit spinal structural damage. The findings of the publication are discussed within the context of the time at the publication and new developments.

The origin of alpine farming: A review of archaeological, linguistic and archaeobotanical studies in the Alps

Alpine farming and pasturing at high altitudes in the Alps has created one of the most species-rich and diverse landscapes in Europe. In order to fully understand, appreciate and protect these habitats it is essential to learn about their history and origin. Until the present day, alpine farming provides essential additional food sources for livestock of farmers in the alpine valleys. Based on written sources, historians are able to track alpine farming back to the Middle Ages. Other approaches from different fields in science, however, can look back even further in search of evidence for alpine farming. This interdisciplinary literature review therefore aims to summarize the scientific work that has been done in different fields of science such as Archaeology, Palynology, Pedoanthracology and linguistic research in order to address the question of the beginning of alpine farming in the Alps. With the discovery of remains from alpine dairy huts, archaeological studies show that there is definite proof of alpine farming beginning in the Bronze Age (2200–800 BC) in different parts of the Alps. Archaeological and palynological data as well as linguistic findings from many different studies and study areas arrived at the same conclusions. Palynological studies found indicators for high-altitude pasture use even earlier, beginning from 4500 BC. The exact type or intensity of this pasture use though remains unclear. In order to confirm these findings, more archaeological research of these areas would be necessary.

Can the Direct Effect of Aerosols Improve Subseasonal Predictability?

The fact that aerosols are important players in Earth’s radiation balance is well accepted by the scientific community. Several studies have shown the importance of characterizing aerosols in order to constrain surface radiative fluxes and temperature in climate runs. In numerical weather prediction, however, there has not been definite proof that interactive aerosol schemes are needed to improve the forecast. Climatologies are instead used that allow for computational efficiency and reasonable accuracy. At the monthly to subseasonal range, it is still worth investigating whether aerosol variability could afford some predictability, considering that it is likely that persisting aerosol biases might manifest themselves more over time scales of weeks to months and create a nonnegligible forcing. This paper explores this hypothesis using the ECMWF’s Ensemble Prediction System for subseasonal prediction with interactive prognostic aerosols. Four experiments are conducted with the aim of comparing the monthly prediction by the default system, which uses aerosol climatologies, with the prediction using radiatively interactive aerosols. Only the direct aerosol effect is considered. Twelve years of reforecasts with 50 ensemble members are analyzed on the monthly scale. Results indicate that the interactive aerosols have the capability of improving the subseasonal prediction at the monthly scales for the spring/summer season. It is hypothesized that this is due to the aerosol variability connected to the different phases of the Madden–Julian oscillation, particularly that of dust and carbonaceous aerosols. The degree of improvement depends crucially on the aerosol initialization. More work is required to fully assess the potential of interactive aerosols to increase predictability at the subseasonal scales.

2 Clement

The text of 2 Clement has only survived in three manuscripts. The oldest one, the Codex Alexandrinus (A), ends at 12.5, the complete Greek text can be found in the Codex Hierosolymitanus (H), which dates back to 1056 ce. The third version can be found in a Syrian translation from 1170 ce. In all three documents, 2 Clem has come down to us in connection with 1 Clem, although it is not a letter but a sermon, more precisely: it is a hortatory address. The whole text deals with parenesis. Chapters 1–3 are dominated by Christological argumentation, in chapters 4–18 the eschatological argumentation has priority. Chapters 19f are a secondary supplement, which probably served as an introduction to chapters 1–18, the original sermon. This can be concluded from linguistic and factual aspects. The author is unknown. He is definitely not the author of 1 Clem, which can be concluded from numerous linguistic differences as well as from different theological views. The assumed opponents are most probably to be found in the context of the emergent Gnosticism, but the author does not focus on a direct confrontation with these opponents. Possible places of origin are Rome, Corinth, Antioch, and Alexandria. In the recent research, Alexandria seems to gain more and more acceptance, based especially on the fact that the author is familiar with Egyptian traditions of his time. It is highly probable that the text was written around 150 ce, and 2 Clem shows that the author is acquainted with the traditions of the Old and New Testaments. Concerning Christian traditions, it can be assumed that he was familiar with the Gospels of Matthew and Luke, but there is no definite proof of this. In addition to texts, which can later be found in canonical documents, 2 Clem also makes use of apocryphal traditions or texts. The formation of the canon was not yet finished at this time. The basic aim of 2 Clem is parenesis. The focus is on the goal to organize life in Christian faith appropriately. Theological topics are oriented toward this goal. God is a creator and a saviour. In his Christology, the author holds the opinion that Jesus is preexistent. It may strike us as remarkable that the author, who is obviously anti-Gnostic, does not estimate pneumatology as important to his audience. The connection between Christ and the church is expressed with the help of syzygies; and interestingly, in this connection the author pleads for the preexistence of the church. In his eschatology, the future version is predominant. Realized eschatological statements can only be found implicitly.

Should the Dose of Antiplatelet Drugs Be Adjusted for Body Weight? The Example of Vorapaxar

Background: In contrast to the vast majority of pharmaceuticals on the market, antiplatelet agents are widely prescribed in a uniform, ‘one size fits all' manner, without conventional dose adjustments. However, strong evidence yielded from clinical trials repeatedly suggests that patients with a low body weight (LBW), the elderly and those with renal or hepatic impairment may benefit from reduced doses, while younger, heavier patients, males and diabetics may benefit from a dose escalation. Vorapaxar, a thrombin protease-activated receptor-1 inhibitor, has been tested in the TRA2P and TRACER clinical trials, but its efficacy and safety in patients with a LBW is unclear. Objective: To determine the impact of LBW on primary end point rates (PER) and bleeding risk after vorapaxar, as yielded from the TRA2P and TRACER secondary FDA review. Results: The LBW (<60 kg) groups in TRA2P (n = 1,852; 7%) and TRACER (n = 1,046; 8%) were small. However, the PER repeatedly suggested inferiority of vorapaxar over placebo in both the successful TRA2P study (10.6 vs. 8.4%; p = 0.012) and the failed TRACER study (19.3 vs. 18.2%; p = not significant). In TRA2P, the PER monotonically escalated with increasing weight for placebo, while those in the vorapaxar arm formed a flat U- or J-shaped distribution across the weight quintiles. In TRACER, the PER by weight quintile appear much higher, but also more random than in TRA2P. The bleeding rates in TRA2P were higher for the 2 lowest-weight quintiles with both placebo and vorapaxar. In TRACER, bleeding rates were more than doubled when compared to TRA2P, and they varied little by weight quintile, with a slight decrease for the heaviest patients in the placebo population and being the highest in the 2 lowest-weight quintiles after vorapaxar. Conclusion: The FDA analyses revealed no definite proof that LBW is associated with reduced efficacy of vorapaxar. While these data are striking, they can be explained by better outcomes in LBW placebo patients already sufficiently treated with dual-antiplatelet therapy. In contrast to efficacy, both TRA2P and TRACER definitely suggest that bleeding rates after vorapaxar are higher in patients with LBW. Dose adjustment for antiplatelet agents may soon become a reality.