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.

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.

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 $\nu$.

The obtention of simple formulae for the Lebesgue constants arising in the the classical Fourier series approximation is presented. Both even and odd cases are treated enlarging Fejér result. Asymptotic formulae are also obtained.

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.

This work presents a new type of chaotic dynamical system originated from Dirichlet Eta function. Although the exposition describes only a bidimensional instance, it is certainly possible to construct higher dimensional such models. The novelty resides in the multitude of interesting characteristics concentrated in only one kind of dynamical system, mainly because of its simple abstract definition - it provokes dissipative, chaotic and itinerant behavior, and its paths typically display nonrecurring quasi-attractors and associate regions of ruin. In addition, it is discrete and nonautonomous, being additively driven by a simple real vector sequence. Also, the defining formulas have parameters which permit to activate bifurcatory effects, mainly related to setting the number of quasi-attractors and overall geometric complexity and diversity. In this fashion, it holds a certain similarity with classical itinerant chaotic systems, but it is a completely different object. One of its parameters is very influential and may be intuitively compared to a type of "potential energy", capable of changing the duration of sojourns etc. In this fashion, there is a potential opportunity to employ artificial inference techniques for designing specific structures on paths. Of course, the other parameters are also relevant, but not necessarily in the control of the number of quasi-attractors and the permanence periods near them. It is important to state that the described system was found during a previous investigation about certain features of the Riemann hypothesis, using the Dirichlet Eta function instead of the Riemann Zeta function - the components of the system under study here were part of certain series whose convergence had to be established, and that scenario produced this very positive side effect. The text also suggests an atypical, but important, potential application in philosophy: the representation of the Hindu concepts of Samsara and Moksha.

This work uses a new approach to investigate the Riemann hypothesis, drawing conclusions about its trueness. It is based on the more general method presented in a very recent publication, by the same author, showing in detail how to approach that very well-known and interesting problem. This is achieved by means of the theory of dynamical systems (Poincaré index associated to equilibria of 2-dimensional systems) and the study of the zeros of the Dirichlet eta function, defined by a Dirichlet series. By using the well-known fact that the zeros of the eta function include all zeros of the Riemann zeta function in the (open) critical strip, excluded the critical line, ((0,1/2) ∪ (1/2,1)) × (−∞,∞), the development proceeds using only the eta function. In addition, the open and simply connected region (1/2,1) × (0,∞) is used along the text, taking into account the symmetries of zeros of the functions under analysis in the critical strip. The basic line of proof is to find the mathematical expression for the Poincaré index of the vector field associated to the eta function, assuming the existence of a zero of the eta function outside the critical line (in (1/2,1) × (0,∞)), and investigating the resulting unfoldings. Eventually, an inconsistency occurs and the proof ends by contradiction.

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.

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.

Big data quality is all about ensuring that the huge amounts of data are accurate, reliable, and useful. It is important because decisions made based on this data can affect many things, like business strategies, customer experiences, and even public policies. Many researchers are studying data quality in the context of big data. This study aims to analyze the research trends and patterns within the domain of big data quality. This method reveals hidden themes in large datasets, helping to understand data quality better. The findings show that Latent Dirichlet Allocation (LDA) is effective for topic modeling in big data, highlighting data quality challenges and opportunities. The research identified key trends and effective assessment methods. For researchers, using LDA in a systematic review helps summarize existing studies and find gaps, guiding future research. Practitioners and future researchers get useful direction for managing data quality and improving efficiency and decisions in various industries.

In this paper, we classify the function space of all real-valued functions on R denoted as F(R,R) into 28 distinct blocks. Each block contains elements that share common features in terms of the cardinality of their sets of continuity and differentiability. Alongside this classification, we introduce the concept of the Connection, which reveals a special relationship structure between four well-known real-valued functions in real analysis: the Cantor function, Dirichlet function, the Thomae function, and the Weierstrass function. Despite the significance of this field, several perspectives remain unexplored.

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.

Topic discovery is finding the main idea of large amounts of textual data. It indicates the recurring topics in the documents, allowing an overview of the texts. Current topic discovery models receive the texts, with or without pre-processing of Natural Language Processing. The processing consists of stopwords removal, text cleaning and normalization (lowercase conversion). A topic discovery model that receives texts with or without processing generates general topics since the input data is many uncategorized texts. The general topics do not offer a detailed overview of the input texts, and manual text categorization is a time-consuming and tedious task. Accordingly, it is necessary to integrate an automatic text classification task in the topic discovery process to obtain specific topics with their top words that contain relevant relationships based on belonging to a class. Text classification performs a word analysis that makes up a document to decide what class or category is being identified; then, integrating the text classification before a topic discovery process will provide latent topics depicted by top words with a high coherence in each topic based on the previously obtained classes. Therefore, this paper exposes a approach that integrates text classification into topic discovery from large amounts of English textual data, such as 20-Newsgroup and Reuters corpora. The text classification is accomplished with a Convolutional Neural Network(CNN) incorporating three embedding models based on semantic relationships. The topic discovery over categorized texts is realized with Latent Dirichlet Analysis(LDA), Probabilistic Latent Semantic Analysis(PLSA), and Latent Semantic Analysis(LSA) algorithms. An evaluation process was performed based on the normalized topic coherence metric. The 20-Newsgroup corpus was classified, and twenty topics with ten top words were discovered for each class, obtaining 0.1723 normalized topic coherence when applying LDA, 0.1622 with LSA, and 0.1716 with PLSA. The Reuters corpus was also classified, obtaining 0.1441 normalized topic coherence when applying the LDA algorithm to obtain 20 topics for each class.

This paper introduces a new adaptation of the enthalpic lattice Boltzmann method (LBM) tailored to simulate the intricate dynamics of conjugate heat transfer in non-homogeneous media, showcasing considerable potential for enhancing heat exchanger designs. This approach expands upon the conventional enthalpic LBM framework by seamlessly incorporating tailored source terms, enabling precise representation of local and transient variations in the medium’s thermophysical properties. The resulting LBM model comprehensively captures the energy conservation equation, with exceptions made only for flow compressibility and viscous dissipation, thereby significantly augmenting its versatility in analyzing heat exchanger configurations. Verification tests, comprising assessments of both transient and steady-state solutions, unequivocally validate the accuracy of the proposed model, further bolstering its reliability for analyzing heat exchange processes within intricate heat exchanger geometries. To demonstrate its efficacy, simulations of conjugate heat transfer processes involving fluid natural convection within structured cavities were performed, imposing both Dirichlet and Neumann boundary conditions to emulate scenarios encountered in practical heat exchanger operations. The findings yield compelling insights, particularly in the transient regime, revealing the beneficial impact of structured cavities on enhancing heat transfer processes. These insights inform potential design optimizations for heat exchangers. The results emphasize the potential of the modified enthalpic LBM approach to elucidate complex heat transfer phenomena in non-homogeneous media and structured geometries, offering valuable implications for heat exchanger engineering and optimization.

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.

In the paper, we prove a joint limit theorem in terms of weak convergence of probability measures on $\mathbb{C}^2$ defined by means of the Epstein ζ(s;Q) and Hurwitz ζ(s,α) zeta-functions. The limit measure in the theorem is explicitly given. For this, some restrictions on the matrix Q and parameter α are required. The theorem obtained extends and generalizes Bohr-Jessen’s results characterising asymptotic behaviour of the Riemann zeta-function.

The purpose of this study is to analyze how global research cooperation patterns in the secondary battery industry have changed over recent years and to identify the evolution in the focus of research. To this end, network analysis was performed using the nationality information of the authors of a 10-year multinational joint research paper related to lithium-ion batteries. Furthermore, keyword analysis and topic modeling were performed using the abstract data from the study. The results of this study confirm that some countries that are not well-known in the field, such as Australia, Spain, and France, showed high centrality, compared with the level of cooperation scale. Additionally, six research topics were identified. According to a comparison over the first half of the decade, no difference was observed in the appearance of keywords indicating high energy density and conductivity with lithium, a key mineral. Keyword distribution was high for topics like battery charging and discharging in the first half of the decade, and for next-generation battery materials, such as solid electrolytes, lithium metal anodes, and lithium-sulfur batteries in the second. These results provide insights into the establishment of R&D cooperation strategies by countries and pre-planning by companies in the battery industry.

The growing adoption of electric propulsion systems in Maritime Autonomous Surface Ships (MASS) necessitates advancements in shipboard electronics for safe, efficient, and reliable operation. These advancements are crucial for tasks like real-time sensor data processing, control algorithms for autonomous navigation, and robust decision-making capabilities. This study investigates research trends in MASS using bibliographic analysis to inform policy and future research directions in this evolving field. We analyze 3,363 MASS-related articles from the Web of Science database, employing co-occurrence word analysis and latent Dirichlet allocation (LDA) topic modeling. Findings reveal a rapidly growing field dominated by image recognition research. Keywords like "datum," "image," and "detection" suggest a focus on collecting and analyzing marine data, particularly with deep learning for synthetic aperture radar imagery. LDA confirms this, with "Image analysis and classification research" as the leading topic. The study also identifies national and organizational leaders in MASS research. However, research on Arctic routes lags behind other areas. This work provides valuable insights for policymakers and researchers, promoting a deeper understanding of MASS and informing future policy and research agendas for the integration of electric propulsion systems in the maritime industry.

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.

Signal approximation, is the basic idea of signal processing and mathematical analysis where complex signals can be represented with simpler ones. This paper explores unifying two mathematical fundamentals, Dirichlet-type problems, and hyperbolic series identities, into signal approximation. In this paper, we address a well-known classic topic in mathematical analysis - the Dirichlet problem - to improve (the approximation of) signal using Fourier series. We also discuss using hyperbolic series representations to increase signal modeling flexibility. In this way, we hope to create new links that span some distance along this dimension and result in a somewhat theoretical view of signal behavior - present with the mathematical rigor needed for speculation but absent from an understanding capable of appropriately handling real-world complexity. This paper provides a theoretical investigation of the mechanics at play and reveals by analogy how these mathematical tools can potentially advance signal processing techniques and understanding. This paper illustrates Fourier series and fuzzy logic methods by two case studies on signal processing, satellite communication and financial risk measurement. The first case study performs signal approximation using the Fourier series and hyperbolic function identities to provide insight into how signals behave and are modeled. The second case study deals with applying fuzzy logic for risk evaluation of investment portfolios, emphasizing the utility in evasion systems solving uncertainty and imprecision problems related to financial decisions.

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.

Fractional differential systems have received much attention in recent decades likely due to its powerful ability in modeling memory processes, which are mostly observed in real world. Fractional models have been widely investigated and applied in many fields such as physics, biochemistry, electrical engineering, continuum and statistical mechanics. It is shown by many researchers that fractional derivatives can provide more accurate models than integer order derivatives do. Obviously, bang-bang property is of a significant importance in optimal control theory. In this paper a Bang-Bang property and time-optimal control problem for time-Fractional Differential System (FDS) is under consideration. First we formulate our problem and we prove the existence theorem. We state and prove the bang-bang theorem. Finally we give the optimality conditions that characterized the optimal control. Some illustrate applications are given to clarify our results.

This research article provides an unconditional proof of an inequality proposed by \textit{Srinivasa Ramanujan} involving the Prime Counting Function $\pi(x)$, \begin{align*} (\pi(x))^{2}<\frac{ex}{\log x}\pi\left(\frac{x}{e}\right) \end{align*} for every real $x\geq \exp(547)$, using specific order estimates of the \textit{Mertens Function}, $M(x)$. The proof primarily hinges upon investigating the underlying relation between $M(x)$ and the \textit{Second Chebyshev Function}, $\psi(x)$, in addition to applying the meromorphic properties of the \textit{Riemann Zeta Function}, $\zeta(s)$ with an intention of deriving an improved approximation for $\pi(x)$.

The core motivation for this study emanates from Voronin S.'s seminal work on the universality of the zeta function. This universality positions the zeta function as a pivotal element in the conceptual framework of global artificial intelligence, suggesting its potential as a foundational component. A critical barrier to harnessing this promising universality has been the absence of efficient algorithms capable of actualizing this concept.