Erdos Conjecture I.

In this short paper I show how it is related to other famous unsolved problems in prime number theory. In order to do this, I formulate the main hypothetical result of this paper - a useful upper bound conjecture (Conjecture 3.), describing one aspect of the distribution of primes in various special forms, paying a brief attention to Fermat, Mersenne, Fibonacci, Lucas and Smarandache sequences, and I debate some side effects of the most surprising results it implies. At the end I also give connections of the questions discussed to other important areas of prime number theory, such as topics from the theory of distribution of primes in denser sequences, and along the way I mention some further conjectures of Erdos that have relevant applications there

New zero free regions for the derivatives of the Riemann zeta function

The main aim of this paper is twofold. First we generalize, in a novel way, most of the known non-vanishing results for the derivatives of the Riemann zeta function by establishing the existence of an infinite sequence of regions in the right half-plane where these derivatives cannot have any zeros; and then, in the rare regions of the complex plane that do contain zeros of the k-th derivative of the zeta function, we describe a unexpected phenomenon, which implies great regularities in their zero distributions. In particular, we prove sharp estimates for the number of zeros in each of these new critical strips, and we explain how they converge, in a very precise, periodic fashion, to their central, critical lines, as k increases. This not only shows that the zeros are not randomly scattered to the right of the line Re(s)=1, but that, in many respects, their two-dimensional distribution eventually becomes much simpler and more predictable than the one-dimensional behavior of the zeros of the zeta function on the line Re(s)=1/2

Compositions with the Euler and Carmichael Functions

http://www.math.missouri.edu/~bbanks/papers/index.htmlLet ' and _ be the Euler and Carmichael functions, respectively. In this paper, we establish lower and upper bounds for the number of positive integers n ≤ x such that '(_(n)) = _('(n)). We also study the normal order of the function '(_(n))/_('(n))

Values of Arithmetical Functions Equal to a Sum of Two Squares

http://www.math.missouri.edu/~bbanks/papers/index.htmlLet '(n) denote the Euler function. In this paper, we determine the order of growth for the number of positive integers n ≤ x for which '(n) is the sum of two square numbers. We also obtain similar results for the Dedekind function (n) and the sum of divisors function _(n)

On the Prime Number Lemma of Selberg

The key result needed in almost all elementary proofs of the Prime Number Theorem is a prime number lemma proved by Atle Selberg in 1948.Without restricting ourselves to purely elementary techniques we show how the error term in Selberg’s fundamental lemma relates to the error term in the Prime Number Theorem. In spite of all the interest in this topic over the last sixty years this particular question seems to have been overlooked in the past

Bounds for Recurrences on Ranked Posets

This note considers an extension of the concept of linear recurrence to recurrences on ranked posets. Some results on growth rates in the linear case are then extended to this generalized scenario. The work is motivated by recent results on multi-dimensional recurrences which have had applications for obtaining bounds for complex multidimensional generating functions. Some further connections to Möbius functions for binary relations and inverses of {0, 1} triangular matrices are also discussed

The Tumor Coagulome as a Transcriptional Target and a Potential Effector of Glucocorticoids in Human Cancers

Background: The coagulome, defined as the repertoire of genes that locally regulate coagulation and fibrinolysis, is a key determinant of vascular thromboembolic complications of cancer. In addition to vascular complications, the coagulome may also regulate the tumor microenvironment (TME). Glucocorticoids are key hormones that mediate cellular responses to various stresses and exert anti-inflammatory effects. We addressed the effects of glucocorticoids on the coagulome of human tumors by investigating interactions with Oral Squamous Cell Carcinoma, Lung Adenocarcinoma, and Pancreatic Adenocarcinoma tumor types. Methods: We analyzed the regulation of three essential coagulome components, i.e., the tissue factor (TF), urokinase-type plasminogen activator (uPA), and plasminogen activator inhibitor-1 (PAI-1) in cancer cell lines exposed to specific agonists of the glucocorticoid receptor (GR) (dexamethasone and hydrocortisone). We used QPCR, immunoblots, small-interfering RNA, Chromatin immunoprecipitation sequencing (ChIPseq) and genomic data from whole tumor and single-cell analyses. Results: Glucocorticoids modulate the coagulome of cancer cells through a combination of indirect and direct transcriptional effects. Dexamethasone directly increased PAI-1 expression in a GR-dependent manner. We confirmed the relevance of these findings in human tumors, where high GR activity/high SERPINE1 expression corresponded to a TME enriched in active fibroblasts and with a high TGF-β response. Conclusion: The transcriptional regulation of the coagulome by glucocorticoids that we report may have vascular consequences and account for some of the effects of glucocorticoids on the TME.</p

КЛІМАТИЧНІ ЗМІНИ ТА ЕКОНОМІЧНА ЕФЕКТИВНІСТЬ АГРАРНОГО ВИРОБНИЦТВА В СТЕПОВІЙ ЗОНІ

Relevance of research. Due to ongoing climate change, almost the entire territory of the Steppe of Ukraine by annual humidity factor belongs to the dry and very dry zones, the relative area of ​​which has increased by 13.2% of the total area of ​​the country compared to the 1960-1990s. At the same time, for today in Ukraine only about 500 thousand hectares are actually irrigated, that is 19% of the potential area. Purpose of research. To determine the patterns and trends of climate change in the western part of the dry Steppe of Ukraine and analyze the economic indicators of production activity in the region as to the variable weather conditions. Research methodology. Climate change was estimated on the basis of Climate Water Balance (CWB) and Hydrothermal Coefficient (HTC) values. The analysis of the economic efficiency of agricultural production was carried out by analyzing the statistical data for Odessa region and for the chosen agricultural enterprise. Research results and conclusions. The use of significant heat supply in the dry Steppe zone is limited by insufficient water supply conditions. In the years of 1991-2016&nbsp; the average annual rainfall was 480 mm and since the early 2000s there has been a slight increase in that. However, even having 500-550 mm of average annual rainfall that has been observed over the past five years, it is not enough for providing high-yield agricultural production. High thermal regime couses high evaporation that in turn, leads to water supply deficit, which at the end of the growing season amounts to 336-436 mm. According to the HTC index in the region 80% of cases show severe and moderately arid vegetation conditions. At the same time, irrigation area in the region has decreased to a minimum, which has led to the domination of winter cereals and sunflower in the cropping system. Under variable weather conditions, winter wheat yields ranged from 19.4 to 38.5 c/ ha (31.4 c/ha on average) and sunflower - from 12.2 to 21.4 c/ha (17.4 c/ha on average), winter rape - from 13.1 to 20.9 c/ha (18.2 c/ha). It was proved a close direct relationship between the sale price of products of all studied crops and their cost price, as well as the inverse relationship of these indicators with the crop yield. The profitability of winter wheat from 2011 till 2016 ranged from 17 to 153 USD/ha with an average value of 86 USD/ha, winter rape - from 39 to 273 USD/ha with an average value of 166 USD&nbsp; ha and sunflower - from 116 to 315 USD/ha with an average value of 192 USD/ha. Corn and soybeans have proven to be unprofitable in some years, which obviously explains rather small areas under these crops in the region. Above mentioned demonstrates the high economic instability of agricultural production in changing weather conditions, which is accompanied by significant risks for producers, especially when attracting credits. This situation, in turn, leads to a limited use of intensification means, in particular mineral fertilizers, which promotes agrochemical soil degradation. Under unstable water supply, the magnitude of net profit variation per hectare of arable land in Odessa region is 33-188 USD/ha (111 USD/ha on average). It is possible to increase these indicators by increasing the share of winter rape in the cropping system. With the optimization of the water and air soil regimes as well as crop rotation factor, the profitability of agricultural production in the region can be increased up to 580-600 USD/ha. Similar results were obtained after analyzing the statistical data from the southern regions within the dry steppe zone.Внаслідок сучасних кліматичних змін майже вся територія Степу України за річним коефіцієнтом зволоження належить до сухої та дуже сухої зон, відносна площа яких збільшилась порівняно з 1960-1990 рр. на 13,2% до загальної площі країни. Водночас по Україні фактично поливається близько 500 тис. га або 19% від вихідної площі. В результаті досліджень встановлено, що використання високого рівня теплових ресурсів в сухостеповій зоні обмежується недостатніми умовами зволоження. За показником ГТК в регіоні у 80% випадків спостерігаються сильно та середньо посушливі умови вегетаційного періоду. Встановлено існування тісної прямої залежності між ціною реалізації продукції всіх досліджуваних культур та їх собівартістю, а також зворотної залежності цих показників із врожайністю посівів. Прибутковість за 2011-2016 рр. коливалася: для пшениці озимої в межах 17-153 у.о./га із середнім значенням 86 у.о./га,&nbsp; ріпаку озимого&nbsp; - 39-273 у.о./га із середнім значенням 166 у.о./га, та соняшнику – 116-315 у.о./га із середнім значенням 192 у.о./га. Кукурудза і соя в окремі роки виявилися збитковими, що очевидно і визначає незначні площі їх посівів в регіоні. Це свідчить про високу&nbsp; економічну нестабільність виробничої діяльності у змінних погодних умовах, що супроводжується значними ризиками для виробників. Загалом, за нестабільних умов зволоження&nbsp; амплітуда коливання чистого прибутку з одного гектара ріллі в Одеській області становить 33-188 у.о./га при середньому рівні 111 у.о./га. Дещо підвищити ці показники можна за збільшення частки в структурі посівних площ ріпака озимого. За оптимізації водно-повітряного режиму ґрунту та сівозмінного фактора прибутковість аграрного виробництва в регіоні можна довести до 580-600 у.о./га. Аналогічні показники отримано в результаті аналізу статистичних даних південних областей в межах сухостепової зони

Does Local Application of Strontium Increase Osteogenesis and Biomaterial Osteointegration in Osteoporotic and Other Bone Tissue Conditions : Review of Literature

Osteoporosis and other pathological bone conditions can impair bone regeneration properties, consuming in increased morbidity and decreased quality of life. Changes of bone healing can result in poor osteointegration and surgical failures if implants are used. To overcome and facilitate bone regeneration, more attempts are made to develop an ideal synthetic scaffold with better biocompatibility, osteoconductivity, bioactivity, osteoinductivity and interconnected porosity. It is considered that strontium, being similar to calcium, can be incorporated into the mineral phase of the bone remodeling. This quality had led strontium to be used as an osteoporotic medication to improve quality of bone and to reduce the risk of bone fractures. Also local application of strontium has been widely used within different biomaterials in tissue engineering researches. In this review authors wanted to provide an overview about strontium, its mechanisms of action in bone tissue and initiated changes of bone remodeling within biomaterials.publishersversionPeer reviewe