Euler’s product formula over the primes and Euler’s zeta function equate to enshrine the Fundamental Theorem of Arithmetic that every integer > 1 is the product of a unique set of primes. The product formula has no zero, and with a domain ≤1 Euler’s zeta diverges. Dirichlet’s eta function η(s), negates alternate terms of zeta, permitting convergence when s∈C and Re(s) < 1, and its non-trivial zeros {ρ}, have a deep relationship with the distribution of the primes. The Riemann Hypothesis is that all the non-trivial zeros have Re(ρ) = 1/2. This work examines the symmetries in a partial Euler’s zeta series with a complex domain equating it to the difference between two finite vector series whose matched terms have mirror-image arguments, but whose magnitudes differ when Re(s) ≠ 1/2. Analytical continuation generates a modified eta series η_l(s), in which every l^th term is multiplied by (1-l). If the integer l is appropriately determined by the Im(s), similar paired finite vector series have a difference that closely follows η_l (s) and their terminal vectors intersect in a unique way permitting zeros only when Re(s) = 1/2. Furthermore, those vectors tracking the derivatives of the series, have a special relationship permitting zeros of the differential only when Re(s) > 1/2.

It is well known that the primes and prime powers have a deep relationship with the nontrivial zeros of Riemann’s zeta function. This is a reciprocal relationship. The zeros and the primes are encoded in each other and are reciprocally recoverable. Riemann’s zeta is an extended or continued version of Euler’s zeta function which in turn equates with Euler’s product formula over the primes. This paper shows that the zeros of the converging Dirichlet or Catalan beta function, which requires no continuation to be valid in the critical strip, can be easily determined. The imaginary parts of these zeros have a deep and reciprocal relationship with the odd primes and odd prime powers. This relationship separates the odd primes into those having either 1 or 3 as a remainder after division by 4. The vector pathway of the beta function is such that the real part of its zeros has to be a half.

In this part of the series of two papers, we extend the theorems discussed in part I for infinite series. We then use these theorems to develop distinct novel results involving the Hurwitz zeta function, Riemann zeta function, Polylogarithm and Fibonacci numbers. In continuation, we obtain some zeros of the newly developed zeta functions and explain their behaviour using plots in complex plane. Furthermore, we provide particular cases for the theorems and corollaries which show that our results generalise the currently available functions and series such as the Riemann zeta function and the geometric series. Finally, we provide four miscellaneous examples to showcase the vast scope of the developed theorems.

In this work, we explore the extreme searching of multidimensional functions by natural gradient descent based on Dirichlet and generalized Dirichlet distributions. The natural gradient is based on describing multidimensional surface with probability distributions, which allows us to reduce changing the accuracy of gradient and step-size. In this article, we propose an algorithm of natural gradient descent based on Dirichlet and generalized Dirichlet distributions. We demonstrate that the natural gradient descent with step-size adaptation with Dirichlet and generalized Dirichlet distributions has higher accuracy and does not take a large number of iterations for minimizing test functions than gradient descent and Adam.

We compute explicitly traces of one-dimensional diffusion processes. The obtained trace forms can be regarded as Dirichlet forms on graphs. Then we discuss conditions ensuring the trace forms to be conservative. Finally, the obtained results are applied to the one-dimensional diffusion related to the Bessel’s process of order $\nu$.

In a series of recent works the relevance of gravitational boundary degrees of freedom and their dynamics in gravity quantization and black hole information has been explored. In this work we further the progress by keenly focusing on the boundary degrees of freedom as the origin of black hole entropy. Wald's entropy formula is scrutinized, and the reason that the Wald's formula correctly captures the entropy of a black hole is examined. Afterwards, limitations of the Wald's method are discussed; a coherent view of entropy based on boundary dynamics is presented. The discrepancy observed in the literature between holographic and Wald's entropies is addressed. We generalize the entropy definition so as to handle a time-dependent black hole. Large gauge symmetry plays a pivotal role. Non-Dirichlet boundary conditions and gravitational analogues of Coleman-De Luccia bounce solutions are central in identifying the microstates and differentiating the origins of entropies associated with different classes of solutions. The result in the present work leads to a view that black hole entropy is entanglement entropy in a thermodynamic setup.

In the field of industrial process monitoring, more and more interest is being shown in specific process categories. These include time-varying processes, that is, those processes whereby the response one receives as output from the system depends on when the input signal is sent into it. There are many reasons for this process variability and such contexts are not always analyzed with this operational characteristic at their core. At the same time, interest in certain categories of techniques is also becoming more prominent, to meet certain application needs. Among these, clustering and unsupervised techniques in general are gaining ground. This is largely due to the difficulty of finding fault data with which to train, for example, supervised models. On the other hand, the clustering technique, on which this contribution focuses, also makes it possible to compensate for the lack of complete knowledge of the structure of the process itself. With these two considerations in mind, this contribution proposes a literature review on the topic of clustering applied in time-varying contexts, in the maintenance field. The aim is to present an overview of the main fields of study, the role of clustering in this context and the main clustering techniques used.

Study question: What are the current trends of research in Human Assisted Reproduction around the world? Summary answer: USA is the leading country, followed by the UK, China, France and Italy. The largest research area is “laboratory techniques”, although other areas such as “public health”, “quality, ethics and law” and “female factor” are gaining ground worldwide. What is known already: Scientific research, especially in health and medical sciences, aims at addressing specific needs that society (and, especially, patients) perceives as pressing. One of the main challenges for policymakers and research funders alike is therefore to align research priorities to societal needs. We can thus think of research agendas in terms of a demand side (societal needs) and a supply side (research outputs). Research output in Human Assisted Reproduction has expanded in the past years, as indicated by the increasing number of scientific publications in indexed journals in this area. Nevertheless, no map of research related to assisted reproduction has been produced so far, hindering the identification of potential areas of improvement and need. Study design, size, duration: 26,000+ scientific publications (articles, letters, and reviews) on Human Assisted Reproduction produced worldwide between 2005 and 2016 were analyzed. These publications were indexed in PubMed or obtained from reference list of indexed publications included in the analysis.Participants/materials, setting, methods: The corpus of publications was obtained by combining the MeSH terms: “Reproductive techniques”, “Reproductive medicine”, “Reproductive health”, “Fertility”, “Infertility”, and “Germ cells”. Then it was analyzed by means of text mining algorithms (Topic Modeling (TM) based on Latent Dirichlet Allocation (LDA)), in order to obtain the main topics of interest. Finally, these categories were analyzed across world regions and time. Main results and the role of chance: We identified 44 main topics, which were further grouped in 11 macro categories, form larger to smaller: “laboratory techniques”, “male factor”, “quality, ethics and law”, “female factor”, “public health and infectious diseases”, “basic research and genetics”, “pregnancy complications and risks”, “general infertility and ART”, “psychosocial aspects”, “cancer”, and “research methodology”. The USA was the leading country in number of publications, followed by the UK, China, France and Italy. Interestingly, research contents in high income countries is fairly homogeneous across macro-categories, and it is dominated by “laboratory techniques” in Western and Southern Europe, and by “quality, ethics and law” in North America, Australia and New Zealand. In middle income countries we observe that research is mainly performed on “male factor”, and noticeably less on “female factor”. Finally, research on “public health and infectious diseases” predominates in low-income countries. Regarding temporal evolution of research, “laboratory techniques” is the most abundant topic on a yearly basis, and relatively constant over time. However, since production in most of the other categories is increasing, the relative contribution of this research category is actually decreasing. Publication is especially increasing in “public health and infectious diseases” (in all world regions, but especially in low income countries), “quality, ethics and law” (high income countries), and “female factor” (middle income countries). Limitations, reasons for caution: Three main factors might limit the robustness of our work: the textual corpus analyzed is based on abstract and titles, the reproducibility of the stochastic algorithms applied, which may produce slightly differing results at each run, and the interpretation of the topics obtained. Wider implications of the findings: This study should prove beneficial in the design of research strategies and policies that foster the alignment between supply (assisted reproduction research) and demand (society). Study funding/competing interest(s): PTQ-14-06718 of the Spanish MINECO Torres Quevedo programme (FAM).

In the paper, by virtue of some properties for the Riemann zeta function, the author finds a double inequality for the ratio of two consecutive Bernoulli numbers with even indexes and analyzes the approximating accuracy of the double inequality.

Relationships between the Farey sequence and the Riemann hypothesis other than the Franel-Landau theorem are discussed. Whether a function similar to Chebyshev’s second function is square-root close to a line having a slope different from 1 is discussed. The nontrivial zeros of the Riemann zeta function can be used to approximate many functions in analytic number theory. For example, it could be said that the nontrival zeta function zeros and the Möbius function generate in essence the same function - the Mertens function. A different approach is to start with a sequence that is analogous to the nontrivial zeros of the zeta function and follow the same procedure with both this sequence and the nontrivial zeros of the zeta function to generate in essence the same function. A procedure for generating such a function is given.

This paper addresses a variant of the product for the Dirichlet $L$--functions. We propose a completely detailed proof for the truth of the generalized Riemann conjecture for the Dirichlet $L$--functions, which states that the real part of the nontrivial zeros is $1/2$. The Wang and Hardy--Littlewood theorems are also discussed with removing the need for it. The results are applicable to show the truth of the Goldbach's conjecture.

In this paper, the existence of two nontrivial stationary solutions for the nonlinear Gilpin Ayala two species competition model is given by using the mountain pass lemma, and the local stability criterion of the trivial solution is given by using Lyapunov function method. Based on the local stability criterion, we give some suggestions on how to avoid the population extinction. This is, when the population is on the verge of extinction, we should try our best to avoid the diffusion behavior and reduce the diffusion coefficient, otherwise the species are easy to go extinct. Numerical example shows the effectiveness of the proposed method.

Ongoing COVID-19 Pandemic has resulted into massive damage to various platforms of global economy which has caused disruption to human livelihood. Natural Language Processing has been extensively used in different organizations to categorize sentiments, perform recommendation, summarizing information and topic modelling. This research aims to understand the non-medical impact of COVID-19 on global economy by leveraging the natural language processing methodology. This methodology comprises of text classification which includes topic modelling on unstructured COVID-19 media articles dataset provided by Anacode. Like other Natural Language Processing algorithms, Latent Dirichlet allocation (LDA) and Non-negative matrix factorization (NMF) has been proposed to classify the media articles dataset in order to analyze COVID-19 pandemic impacts in the different sectors of global economy. Model Accuracy was examined based on the coherence and perplexity score which came out to be 0.51 and -10.90 using LDA algorithm. Both the LDA and NMF algorithm identified similar prevalent topics that was impacted by COVID-19 pandemic in multiple sectors of economy. Through intertopic distance map visualization produced by LDA algorithm, it can be reciprocated that general industries which includes children schooling, parental care, and family gatherings had the major impact followed by business sector and the financial industry.

The research presents the Methodology of Improving the Accuracy in Text Classification in Light of Modelling the Latent Semantic Relations (LSR). The aim of this Methodology is to find the ways of eliminating the Limitations of Discriminant and Probabilistic methods for LSR revealing and customizing the Text Classification Process to the more accurate recognition of the text tonality. This aim should be achieved by using the knowledge about the text’s Hierarchical Semantic Context in the form of Corpora-based Hierarchical Sentiment Dictionary. The main scientific contribution of this research is the following set of approaches to improve the qualitative characteristics of Text Classification process: combination of the Discriminant and Probabilistic methods allowing to decrease the influences of the Limitations of these methods on the LSR revealing process; considering each document as a complex structure allowing to estimate documents integrally by separated classification of topically completed textual component (paragraphs); taking into account the features of Argumentative type of documents (Reviews) allowing to use the author’s subjective evaluation of text tonality for development the Text Classification methodology. Tonality, expressed by the Review’s author, has a significant, but not critical, effect on the qualitative indicators of Sentiment Recognition.