In this paper, we introduce the bicomplex Tetranacci and Tetranacci-Lucas quaternions. Moreover, we present Binet's formulas, generating functions, and the summation formulas for those bicomplex quaternions.

In this paper, we introduce the generalized Tetranacci quaternions and present some properties of these quaternions.

In this paper, we define Tetranacci and Tetranacci-Lucas matrix sequences and investigate their properties.

This paper deals with developing a new class of quaternions, called hyper-dual Horadam quaternions which are constructed from the quaternions whose components are hyper-dual Horadam numbers. We investigate the algebraic properties of these quaternions.

Here, we describe the bicomplex (p,q)- Fibonacci numbers and the bicomplex (p,q)- Fibonacci quaternions that are based on these numbers and give some of their equations, including the Binet formula, generating function, Catalan, Cassini, d’Ocagne’s identities, and some summation formulas for both of them. Finally, we create a matrix for bicomplex (p,q)- Fibonacci quaternions, and we obtain a determinant of a special matrix that gives the terms of that quaternion.

The aim of this paper is to bring together quaternions and generalized complex numbers. Generalized quaternions with generalized complex number components are expressed and their algebraic structures are examined. Several matrix representations and computational results are introduced. As a crucial part, alternative approach for generalized quaternion matrix with elliptic number entries are developed.

In this paper, inspired by recent articles of G. Anatriello and G. Vincenzi (see \cite{Ana}), and G. Cerda-Morales (see \cite{Cer1,Cer2}), we will introduce the third-order $\overline{h}$-Jacobsthal and third-order $\overline{h}$-Jacobsthal--Lucas sequences and their associated quaternions. The new results that we have obtained extend most of those obtained in \cite{Cer2}.

This article derives the closed form solution to finding the roots of a quadratic equation in SU3.

In this paper, closed forms of the sum formulas $\sum_{k=0}^{n}W_{k}^{2}$ \ for the squares of generalized Tetranacci numbers are presented. We also present the sum formulas $\sum_{k=0}^{n}W_{k+1}W_{k},$ $\sum_{k=0}^{n}W_{k+2}W_{k},$ and $\sum_{k=0}^{n}W_{k+3}W_{k}.$ As special cases, we give summation formulas of the of Tetranacci, Tetranacci-Lucas and some other fourth order linear recurrance sequences.

In this paper, we introduce hybrid numbers with Fibonacci and Lucas hybrid number coefficients. We give the Binet formulas, generating functions, exponential generating functions for these numbers. Then we define an associate matrix for these numbers. In addition, using this matrix, we present two different versions of Cassini identitiy of these numbers.

In this paper, we define and investigate modified p-Leonardo, p-Leonardo-Lucas and p-Leonardo sequences as special cases of the generalized Leonardo sequence. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences. Furthermore, we show that there are close relations between modified p-Leonardo, p-Leonardo-Lucas, p-Leonardo numbers and Fibonacci, Lucas numbers.

In this paper, by using the q-integer, we introduce a new generalization of Fibonacci and Lucas sedenions called q-Fibonacci and q-Lucas sedenions. We present some fundamental properties of these type of sedenions such as Binet formulas, exponential generating fuctions, summation formulas, Catalan's identity, Cassini's identity and d'Ocagne's identity.

Net CO2 emissions and sequestration from European forests are the result of removal and growth of flora. To arrive at aggregated measurements of these processes at a country's level, local observations of increments and harvest rates are up-scaled to national forest areas. Each country releases these statistics through their individual National Forest Inventory using their particular definitions and methodologies. In addition, five international processes deal with the harmonization and comparability of such forest datasets in Europe, namely the IPCC, SOEF, FAOSTAT, HPFFRE, FRA (definitions follow in the article). In this study, we retrieved living biomass dynamics from each of these sources for 27 European Union member states. To demonstrate the reproducibility of our method, we release an open source python package that allows for automated data retrieval and analysis, as new data becomes available. The comparison of the published values shows discrepancies in the magnitude of forest biomass changes for several countries. In some cases, the direction of these changes also differ between sources. The scarcity of the data provided, along with the low spatial resolution, forbids the creation or calibration of a pan-European forest dynamics model, which could ultimately be used to simulate future scenarios and support policy decisions. To attain these goals, an improvement in forest data availability and harmonization is needed.

In the paper, the authors discover several series identities involving the Catalan numbers, the Catalan function, the Riemanian zeta function, and the alternative Hurwitz zeta function.

Integer is either a composite number or a prime number. Therefore, detecting composite numbers is important for solving prime numbers. The study of prime numbers, apart from satisfying human curiosity, can be very important. In this article, the order of composite numbers has been detected. And explained with a simple method and a simple function. And, a method has been developed in which all composite numbers and therefore prime numbers can be determined by using the specified methods, functions and formulas.

Multi-receiver encryption enables a sender to encrypt a message and transmit the ciphertext to a set of authorized users, while no one out this group of authorized users can decrypt the message. Multi-receiver encryption is of great importance in many sectors such as broadcast communication, cloud computing, wireless communications, networking applications, e-voting, lottery, and medical applications. This paper proposes an efficient multi-receiver public key encryption scheme based on Lucas sequences. The results of the computational analysis show that, projected scheme is better against renown attacks and prevailing anonymous multireceiver algorithms

In the paper, the authors find closed forms for derangement numbers in terms of the Hessenberg determinants, discover a recurrence relation of derangement numbers, present a formula for any higher order derivative of the exponential generating function of derangement numbers, and compute some related Hessenberg and tridiagonal determinants.

This paper addresses the evolving landscape of electricity markets in Europe, with a focus on the integration of Renewable Energy Communities as introduced by the Renewable Energy Directive 2018/2001. Residents within a postal code area are highly incentivized to join a community, which enables them to exchange energy among themselves at lower procurement costs. Thereby, energy management systems optimize the operation of respective energy systems, with electrical load forecasting playing a key role. Given that prosumers may switch between communities on a daily basis, electricity demands within these groups will vary, leading to data that is non-stationary, discontinuous as well as non-identical and independently distributed. To encounter this issue, we propose a sophisticated forecasting model that applies federated learning, using informations from various distributed communities to learn domain-invariant features. To achieve this, we initially utilize synthetic electrical load time series at district level and aggregate them to profiles of Renewable Energy Communities with dynamic portfolios. Subsequently, we develop a forecasting model that accounts for the composition of residents of a Renewable Energy Community, adapt data pre-processing in accordance with the time series process, and detail a federated learning algorithm that incorporates weight averaging and data sharing. Following the training of various experimental setups, we ultimately evaluate their effectiveness. The findings suggest that our proposed framework is capable of effectively forecast non-stationary and discontinuous time series and that it can be applied to new, unseen data through the integration of knowledge from multiple sources.

There are many new land use and land cover (LULC) products emerging yet there is still a lack of in-situ data for training, validation, and change detection purposes. The LUCAS (Land Use Cover Area frame Sample) survey is one of the few authoritative in-situ field campaigns, which takes place every three years in European Union member countries. More recently, a study has considered whether citizen science and crowdsourcing could complement LUCAS survey data, e.g., through the FotoQuest Austria mobile app and crowdsourcing campaign. Although the data obtained from the campaign were promising when compared with authoritative LUCAS survey data, there were classes that were not well classified by the citizens, and the photographs submitted through the app were not always of sufficient quality. For this reason, in the latest FotoQuest Go Europe 2018 campaign, several improvements were made to the app to facilitate interaction with the citizens contributing and to improve their accuracy in LULC identification. In addition to extending the locations from Austria to Europe, a change detection component (comparing land cover in 2018 to the 2015 LUCAS photographs) was added, as well as an improved LC decision tree and a near real-time quality assurance system to provide feedback on the distance to the target location, the LULC classes chosen and the quality of the photographs. Another modification was the implementation of a monetary incentive scheme in which users received between 1 to 3 Euros for each successfully completed quest of sufficient quality. The purpose of this paper is to present these new features and to compare the results obtained by the citizens with authoritative LUCAS data from 2018 in terms of LULC and change in LC. We also compared the results between the FotoQuest campaigns in 2015 and 2018 and found a significant improvement in 2018, i.e., a much higher match of LC between FotoQuest Go Europe and LUCAS. Finally, we present the results from a user survey to discuss challenges encountered during the campaign and what further improvements could be made in the future, including better in-app navigation and offline maps, making FotoQuest a model for enabling the collection of large amounts of land cover data at a low cost.

The theory of harmonic based function is discussed here within the framework of umbral operational methods. We derive a number of results based on elementary notions relying on the properties of Gaussian integrals.

The purpose of this work is to describe and explain the characteristics of pedagogy as a science of education, as an object, concepts and research method. The methodology assumed was the review of the history of pedagogy, especially the classics of pedagogy, those who built it with their experiences in laboratory schools. Here are listed the main concepts of pedagogy, its laws and principles; she establishes the opposition that exists between teaching-learning and education as formation of the personality; she also delimits the fields of the pedagogical method, of didactics or teaching-learning, and the methodology of educational labor that organizes the process and the educational context, the regime and the lifestyle, such as those experienced in the "colonies" or "communes" of children and adolescents, organized and directed by Neill and Makarenko. Since the present article does not derive from a primary source or from field work, as an empirical or experimental research, the results constitute the content of the article, as conclusions.

Laser powder bed fusion (LPBF) is an additive manufacturing process employed in many industries, for example for aerospace, automotive and medical applications. In these sectors, mainly nickel-, aluminum- and titanium-based alloys are used. In contrast, the mechanical engineering industry is interested in more wear-resistant steel alloys with higher hardness, both of which can be achieved with a higher carbon content, like in high-speed steels. Since these steels are susceptible to cracking, preheating needs to be applied during processing by LPBF. In a previous study, we applied a base plate preheating temperature of 500 °C for HS6-5-3-8 with 1.3 % carbon content. We were able to manufacture dense (p > 99.9 %) and crack-free parts from HS6-5-3-8 with a hardness > 62 HRC (as built) by LPBF. In this study, we investigate the influence of preheating temperatures up to 600 °C on hardness and microstructure dependent on part height for HS6-5-3-8. The microstructure was studied by light optical microscopy (LOM), scanning electron microscopy (SEM) and electron backscatter diffraction (EBSD). The analysis of hardness and microstructure at different part heights is necessary because state-of-the-art preheating systems induce heat only into the base plate. Consequently, parts are subjected to temperature gradients and different heat treatment effects depending on part height during the LPBF process.

This paper presents a simple drone survey and image processing approach to identifying St. John's wort (Hypericum perforatum) on grazing land in the Hunter Valley NSW, Australia. St John’s wort is an invasive species (weed) that competes with pasture, poisons livestock, can downgrade wool with ‘vegetable fault’, and decreases property values. Identifying the locations of St John wort from the ground can be difficult due to topography, limited access, and/or larger land areas of mixed vegetation. In this study, a drone was used to survey a 174-ha grazing property in the Hunter Valley NSW (Australia). The images were stitched together using commercially available software. A unique Visual Atmospheric Resistance Index (VARI) attribute was identified and used to highlight the presence of St John’s wort in the survey area. These sub-areas were then digitised onto high resolution maps for future planning use by land managers.

(S)-Atenolol ((S)-2-(4-(2-Hydroxy-3-(isopropylamino)propoxy)phenyl)acetamide) has been synthesized in >99% ee with use of Candida antarctica lipase B from Syncozymes (Shanghai, China), in a kinetic resolution of the corresponding racemic chlorohydrin. Deprotonation of the starting phenol building block was base catalyzed. The enantiopurity of the chlorohydrin building block remained unchanged upon subsequent amination. All four steps in the synthesis protocol have been optimized compared to previously reported methods, which makes this new protocol more sustainable and in accordance with green chemistry principles compared to previously reported protocols.

In this paper, we study sums of finite products of Chebyshev polynomials of the first kind and Lucas polynomials and represent each of them in terms of Chebyshev polynomials of all kinds. Here the coefficients involve terminating hypergeometric functions ₂F₁ and these representations are obtained by explicit computations.

If environmental exposures are shown to cause an adverse health outcome, reducing exposure should reduce the disease risk. Links between exposures and outcomes are typically based on ‘associations’ derived from observational studies, and causality may not be clear. Randomised controlled trials to ‘prove’ causality are often not feasible or ethical. Here the history of evidence that tobacco smoking causes lung cancer – in observational studies – is compared to that of low sun exposure and/or low vitamin D status as causal risk factors for the autoimmune disease, multiple sclerosis. Evidence derives from in vitro and animal studies, as well as ecological, case-control and cohort studies, in order of increasing strength. For smoking and lung cancer, the associations are strong, consistent, and biologically plausible – the evidence is coherent or ‘in harmony’. For low sun exposure/vitamin D as risk factors for MS, the evidence is weaker, with smaller effect sizes, but coherent across a range of sources of evidence, and biologically plausible. The association is less direct – smoking is directly toxic and carcinogenic to the lung, but sun exposure/vitamin D modulate the immune system, which in turn may reduce the risk of immune attack on self-proteins in the central nervous system. Opinion about whether there is sufficient evidence to conclude that low sun exposure/vitamin D increase the risk of multiple sclerosis, is divided. General public health advice to receive sufficient sun exposure to avoid vitamin D deficiency (<50nmol/L) should also ensure any benefits for multiple sclerosis.

In this paper, we investigate the generalized Padovan sequences and we deal with, in detail, four special cases, namely, Padovan, Perrin, Padovan-Perrin and modified Padovan sequences. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences.

Number is based in special edges, is introduced. The kind of natural extension from edges toward vertex and the set of vertices in the way that, the final notion is number, is studied. The result is obtained which is about the study on the classes of graphs in the matter of new notions. There is the extended notion about having edge amid two vertices toward having some edges in the word of neighbor and in another stage going into the atmosphere of having consecutive edges in the terminology of path and in the upper vision going on the notion about path with jargon and buzzword of distance as if the minimal vertices has concluded the new notions with the word, number.

New setting is introduced to study “closing numbers” and “super-closing numbers” as optimal-super-resolving number, optimal-super-coloring number and optimal-super-dominating number. In this way, some approaches are applied to get some sets from (Neutrosophic)n-SuperHyperGraph and after that, some ideas are applied to get different types of super-closing numbers which are called by optimal-super-resolving number, optimal-super-coloring number and optimal-super-dominating number. The notion of dual is another new idea which is covered by these notions and results. In the setting of dual, the set of super-vertices is exchanged with the set of super-edges. Thus these results and definitions hold in the setting of dual. Setting of neutrosophic n-SuperHyperGraph is used to get some examples and solutions for two applications which are proposed. Both setting of SuperHyperGraph and neutrosophic n-SuperHyperGraph are simultaneously studied but the results are about the setting of n-SuperHyperGraphs. Setting of neutrosophic n-SuperHyperGraph get some examples where neutrosophic hypergraphs as special case of neutrosophic n-SuperHyperGraph are used. The clarifications use neutrosophic n-SuperHyperGraph and theoretical study is to use n-SuperHyperGraph but these results are also applicable into neutrosophic n-SuperHyperGraph. Special usage from different attributes of neutrosophic n-SuperHyperGraph are appropriate to have open ways to pursue this study. Different types of procedures including optimal-super-set, and optimal-super-number alongside study on the family of (neutrosophic)n-SuperHyperGraph are proposed in this way, some results are obtained. General classes of (neutrosophic)n-SuperHyperGraph are used to obtains these closing numbers and super-closing numbers and the representatives of the optimal-super-coloring sets, optimal-super-dominating sets and optimal-super-resolving sets. Using colors to assign to the super-vertices of n-SuperHyperGraph and characterizing optimal-super-resolving sets and optimal-super-dominating sets are applied. Some questions and problems are posed concerning ways to do further studies on this topic. Using different ways of study on n-SuperHyperGraph to get new results about closing numbers and super-closing numbers alongside sets in the way that some closing numbers super-closing numbers get understandable perspective. Family of n-SuperHyperGraph are studied to investigate about the notions, super-resolving and super-coloring alongside super-dominating in n-SuperHyperGraph. In this way, sets of representatives of optimal-super-colors, optimal-super-resolving sets and optimal-super-dominating sets have key role. Optimal-super sets and optimal-super numbers have key points to get new results but in some cases, there are usages of sets and numbers instead of optimal-super ones. Simultaneously, three notions are applied into (neutrosophic)n-SuperHyperGraph to get sensible results about their structures. Basic familiarities with n-SuperHyperGraph theory and neutrosophic n-SuperHyperGraph theory are proposed for this article.

In the paper, by virtue of the Faá di Bruno formula and two identities for the Bell polynomial of the second kind, the authors find a closed form for the Stirling polynomials in terms of the Stirling numbers of the first and second kinds.

There is now evidence that, based on cytokine profiles, bipolar disorder (BD) is accompanied by simultaneous activation of the immune-inflammatory response system (IRS) and the compensatory immune-regulatory system (CIRS), and that both components may be associated with the staging of illness. Nevertheless, no BD studies have evaluated the IRS/CIRS ratio using CD (cluster of differentiation) molecules expressed by peripheral blood activated T effector (Teff) and T regulatory (Treg) subpopulations. This study examined T cell subsets both before and after ex vivo anti CD3/CD28 stimulation using flow cytometric immunophenotyping in 25 euthymic BD patients and 21 healthy controls as well as human cytomegalovirus (HCMV)-specific IgG antibodies. BD is associated with a significantly lowered frequency of baseline (unstimulated) CD3+CD8+CD71+ and CD4+CD25+FOXP3 and increased CD4+CD25+FOXP3+CD152+ frequencies and with lowered stimulated frequencies of CD3+CD8+CD71+, CD4+CD25+FOXP3+CD152+ and CD4+CD25+FOXP3+GARP cells and, consequently, by an increased stimulated Teff/Treg ratio. Moreover, the number of manic, but not hypomanic or depressive episodes, is significantly and negatively associated with the stimulated proportions of CD3+CD4+CD154+, and CD69+ and CD71+ expression on CD4+ and CD8+ cells, while duration of illness (≥ 10 years) is accompanied by a depleted frequency of stimulated CD152+ Treg, and CD154+ and CD71+ CD4+ T cells. BD and anti-human cytomegalovirus (HCMV) IgG levels significantly interact to decrease the expression of CD4+CD25+FOXP+GARP T phenotypes. In conclusion, BD is characterized by deficits in immune-regulatory functions while the staging of illness is characterized by additional impairments is Teff and Treg activation. HCMV seropositivity may contribute to an immune-risk phenotype associated with BD.

This article presents an in-depth review of Virtual Power Plant (VPP), its organization in the energy system and its prospects in the face of the promising future of the increasing bidirectional complexity exposed in the current scenario. A new paradigm for the operations of modern power distribution and transmission systems requires greater grid flexibility that is accompanying an extensive change in the structure of electricity markets, the fruit of the development of the renewable industry and of the growing photovoltaic systems popularized, but in contrast due to the intermittent nature cause variable uncertainties in the power system. Innovative concepts like VPP are becoming a reality, establishing an efficient and effective mechanism. The objectives and components are described in a comprehensive way, and some of the most important are pointed out and presented in detail to contribute with a description of the energy systems and the implicit research needed for sustainability and resilience in the eminent energy scenario with this technology. In addition, the literature and studies of this technology already indicate a direction of this new tool as a promising solution to manage the uncertainties of the renewable energies.

The article shows materials to the question about algebraic features of the genetic code and about the dictatorial influence of the DNA and RNA molecules on the whole organism. Presented results testify in favor that the genetic code is an algebraic code related with a wide class of algebraic codes, which are a basis of noise-immune coding of information in communication technologies. Structural features of the genetic systems are associated with hypercomplex double (or hyperbolic) numbers and with bisymmetric doubly stochastic matrices. The received results confirm that represented matrix approaches are effective for modeling genetic phenomena and revealing the interconnections of structures of biological bodies at various levels of their organization. This allows one to think that living organisms are algebraically encoded entities where structures of genetic molecules have the dictatorial influence on inherited structures of the whole organism. New described algebraic approaches and results are discussed.

In this paper, we present Binet's formulas, generating functions, and the summation formulas for generalized Pentanacci numbers, and as special cases, we investigate Pentanacci and Pentanacci-Lucas numbers with their properties. Also, we define Gaussian generalized Pentanacci numbers and as special cases, we investigate Gaussian Pentanacci and Gaussian Pentanacci-Lucas numbers with their properties. Moreover, we give some identities for these numbers. Furthermore, we present matrix formulations of generalized Pentanacci numbers and Gaussian generalized Pentanacci numbers.

In this paper, we introduce the partially degenerate Changhee-Genocchi polynomials and numbers and investigated some identities of these polynomials. Furthermore, we investigate some explicit identities and properties of the partially degenerate Changhee-Genocchi arising from the nonlinear dierential equations.

In the paper, the authors survey integral representations of the Catalan numbers and the Catalan–Qi function, discuss equivalent relations between these integral representations, supply alternative and new proofs of several integral representations, collect applications of some integral representations, and present sums of several power series whose coefficients involve the Catalan numbers.

In this age of mass media and, in particular, social media-driven perception of reality, coupling disease and prophylactic opinion dynamics models can provide better insights into disease evolution than using a disease model alone. We develop in this work two disease-opinion dynamics models based on the epidemiology of the new coronavirus disease (COVID-19) and the availability or not of imperfect vaccines. We assume that susceptibility to infection decreases with the level of prophylactic attitude (personal hygiene, social distancing), and changes in prophylactic attitudes of susceptible individuals occur in response to perceived disease prevalence and vaccination coverage and efficacy in the population. We derive and discuss the disease-free equilibriums and reproduction numbers in the introduced models. We further assess the impacts of the distribution of opinions at disease introduction, the ability to detect presymptomatic, asymptomatic and symptomatic positive COVID-19 cases, the behavioural responses to the outbreak and the introduction of vaccination, and the effects of distortions of disease prevalence by public policy and mass media on disease dynamics. The insights highlighted from the proposed models are expected to make informative contributions to public policy in a context of opinion fluxes in response to perceived disease prevalence.

When presenting special relativity, it is customary to single out the so-called paradoxes. One of these paradoxes is the formal occurrence of speeds exceeding the speed of light. An essential part of such paradoxes arises from the incompleteness of the relativistic calculus of velocities. In special relativity, the additive group is used for velocities. However, the use of only group operations imposes artificial restrictions on possible computations. Naive expansion to vector space is usually done by using non-relativistic operations. We propose to consider arithmetic operations in the special theory of relativity in the framework of the Cayley–Klein model for projective spaces. We show that such paradoxes do not arise in the framework of the proposed relativistic extension of algebraic operations.

Elliptic scator algebra is possible in 1+n dimensions, n∈N. It is isomorphic to complex algebra in 1+1 dimensions, when the real part and any one hypercomplex component are considered. It is endowed with two representations: an additive one, where the scator components are represented as a sum; and a polar representation, where the scator components are represented as products of exponentials. Within the scator framework, De Moivre’s formula is generalized to 1+n dimensions in the so called victoria equation. This novel formula is then used to obtain compact expressions for the integer powers of scator elements. A scator in S1+n can be factored into a product of n scators that are geometrically represented as its projections onto n two dimensional planes. A geometric interpretation of scator multiplication in terms of rotations with respect to the scalar axis is expounded. The powers of scators, when the ratio of their director components is a rational number, lie on closed curves. For 1+2 dimensional scators, twisted curves in a three dimensional space are obtained. Collecting previous results, it is possible to evaluate the exponential of a scator element in 1+2 dimensions.

The importance and near ubiquity of the Golden Ratio in disciplines like chemistry and biology is well known, but only recently has it come to light in areas pertaining to the Quantum domain. By using a modified tool-kit of hyper-complex numbers (known as Dimensional Gate Operators) and numerical analysis, we uncover a connection between hyper-dimensional objects, the Fibonacci sequence and that of quasiparticles and excitations. The results show that dimensionality increases in step with the Fibonacci sequence.

Purpose To analyse diagnostic and therapeutic impact of molecular imaging TNM (miTNM) stage obtained with 18F-DCFPyL versus 18F-Fluorocholine in head to head comparison in biochemical recurrence (BCR) prostate cancer (PCa). Material and methods Patients with BCR PCa after radical treatment with previous 18F-Fluorocholine-PET/CT (negative or oligometastatic disease) were derived to 18F-DCFPyL-PET/CT. Patients were classified attending to: grade group, EAU classification, PSA, PSA doubling time (PSAdt) and PSA velocity (PSAvel). The overall detection rate (DR) and miTNM stage according to PROMISE criteria were assessed for both radiotracers and also correlation (Kappa). The influence of PSA and kinetics on both PET/CT (DR and miTNM) and predictive value of unfavourable kinetics on miTNM were determined. Cut-off PSA, PSAdt and PSAvel values able to predict PET/CT results were determined. Change in miTNM and treatment derived of 18F-DCFPyL information with respect to 18F-Fluorocholine was also evaluated. Results We studied 138 patients. 18F-DCFPyL showed a higher DR than 18F-Fluorocholine (64.5% vs 33.3%). 18F-DCFPyL and 18F-Fluorocholine detected T in 33.3% versus 19.6%, N in 27.5% versus 13.8%, M in 30.4% versus 8.7%. Both tracers DR showed significant associations with PSA and PSAvel. Significant association was only found between miTNM and PSA on 18F-Fluorocholine-PET/CT (p=0.033). For 18F-Fluorocholine and 18F-DCFPyL PET/CT a PSAdt cut-off of 4.09 and 5.59 months, respectively, were able to predict M stage. 18F-DCFPyL changed therapeutic management in 40/138 patients. Conclusion 18F-DCFPyL provides a higher DR and superior miTNM staging than 18F-Fluorocholine in restaging BCR, especially with high PSA and unfavourable PSA kinetics, showing a fair agreement to 18F-Fluorocholine.

Prime number-related issues can be viewed from drastically different perspectives by examining the close connections between prime numbers and composite numbers. We think that multiple perspectives are the pillars on the path to solutions so we have created this study. As a result of the study, we proposed two new formulas by presenting three theorems and one proof for each theorem, a total of three proofs. We proved that the formula p · n + p returns a composite number in the first of the theorems, which is the preliminary theorem. Our first theorem except the preliminary theorem is that the formula p · n + p returns all composite numbers, and we proved that too. Finally, we created Theorem II using Theorem I to use in our other work and proved that the formula 2 · n · p + p returns all odd composite numbers, which is Theorem II. Afterward, we presented the similarities of the 2 · n · p + p formula we put forth with another known formula.

In the paper, the authors present unified generalizations for the Bell numbers and polynomials, establish explicit formulas and inversion formulas for these generalizations in terms of the Stirling numbers of the first and second kinds with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem connected with the Stirling numbers of the first and second kinds, construct determinantal and product inequalities for these generalizations with aid of properties of the completely monotonic functions, and derive the logarithmic convexity for the sequence of these generalizations.

A new generalized definition of Mersenne numbers is proposed of the form (a^n - (a-1)^n), called Global Generalized Mersenne numbers, or simply Generalized Mersenne numbers and noted GM_{a,n} where a is the base and n is the exponent, both being positive integers. The properties are investigated for prime exponents n and several theorems on Mersenne numbers regarding their congruence properties are generalized and demonstrated. In particular, it is found that, for any base a, Generalized Mersenne numbers are in general such that (GM_{a,n} - 1) are even and divisible by n, a and (a-1) for any odd prime exponent n and by (a(a-1)+1) for any prime exponent n > 5. The remaining factor is a function of triangular numbers of (a-1), specific to each prime exponent n. Four theorems on Mersenne numbers are generalized and four new theorems are demonstrated, allowing to show first, that (GM_{a,n} - 1) are divisible by 6, and more precisely GM_{a,n} are congruent to 1(mod 12) or 7(mod 12) depending on the congruence of the base a(mod 4); second, that (GM_{a,n} - 1) are divisible by 10 if n == 1(mod 4) and, if n == 3(mod 4), GM_{a,n} == 1(mod 10), or 7(mod 10) or 9(mod 10) depending on the congruence of the base a(mod 5); third, that all factors c_{i} of GM_{a,n} are of the form (2nf_{i}+1) with f_{i} natural integers such that c_{i} is prime itself or the product of primes of the form (2nj+1) with j natural integer; fourth, that for odd prime exponents n, all GM_{a,n} are periodically congruent to either +/-1\(mod 8) or +/-3(mod 8) depending on the congruence of the base a(mod 8); and fifth, that the factors of a composite GM_{a,n} is of the form (2nf_{i}+1) with f_{i} == u(mod 4) and u being either 0, 1, 2 or 3 depending on the congruence of the exponent n(mod 4) and on the congruence of the base a(mod 8). The potential use of Generalized Mersenne primes in cryptography is shortly addressed.

Cognitive linguistics is founded on the assumption that language is a reflection of universal cognitive principles, implying that all cognitive processes share a common origin. This assumption is known as the 'Cognitive Commitment.' The present article examines the sensory schema hypothesis, which traces all forms of expression back to sensations. This theoretical framework integrates the physicality of the body with sensations that underlie human experiences and their linguistic and non-linguistic expressions. By challenging the traditional dichotomy between mind and body, it provides a more integrated understanding.

We consider an $N$ fermion system at low temperature $T$ in which we encounter special particle number values $N_m$ exhibiting special traits. These values arise in focusing attention upon the degree of mixture (DM) of the pertinent quantum states. Given the coupling constant of the Hamiltonian, the DMs stay constant for all $N$-values, but experience sudden jumps at the $N_m$. For a quantum state described by the matrix $\rho$, its purity is expressed by $Tr \rho^2$ and then the degree of mixture is given by $1 - Tr \rho^2$, a quantity that coincides with the entropy $S_q$ for $q=2$. Thus, Tsallis entropy of index two faithfully represents the degree of mixing of a state, that is, it measures the extent to which the state departs from maximal purity . Macroscopic manifestations of the degree of mixing can be observed through various physical quantities. Our present study is closely related to properties of many-fermion systemsn that are usually manipulated at zero temperature. Here we wish to study the subject at finite temperature. Gibbs' ensemble is appealed to. Some interesting insights are thereby gained.

In this paper, we introduce a new class of degenerate Hermite-Fubini numbers and polynomials and investigate some properties of these polynomials. We establish summation formulas of these polynomials by summation techniques series. Furthermore, we derive symmetric identities of degenerate Hermite-Fubini numbers and polynomials by using generating functions.

The main purpose of this paper is to introduce a new class of $q$-Hermite-Fubini numbers and polynomials by combining the $q$-Hermite polynomials and $q$-Fubini polynomials. By using generating functions for these numbers and polynomials, we derive some alternative summation formulas including powers of consecutive $q$-integers. Also, we establish some relationships for $q$-Hermite-Fubini polynomials associated with $q$-Bernoulli polynomials, $q$-Euler polynomials and $q$-Genocchi polynomials and $q$-Stirling numbers of the second kind.

In this paper, we investigate a new $q$-analogue of the higher order degenerate Changhee polynomials and numbers, which are called the Witt-type formula for the $q$-analogue of degenerate Changhee polynomials of order $r$. We can derive some new interesting identities related to the degenerate $(h,q)$-Changhee polynomials and numbers.

In this paper, we investigate a new $q$-analogue of the higher order degenerate Changhee polynomials and numbers, which are called the Witt-type formula for the $q$-analogue of degenerate Changhee polynomials of order $r$. We can derive some new interesting identities related to the degenerate $(h,q)$-Changhee polynomials and numbers.

Bringing toghether mathematical and philosophical ideas related to transcendental numbers, this paper begins with a survey on transcendence and transcendental numbers, it then presents several properties of the transcendental numbers e and π, and then it gives the proof of a new inequality for transcendental numbers. Also, in relationship with these topics, we study solutions to the Yang-Baxter equation from hyperbolic functions and from logical implication.

In the present paper, we introduce (p, q)-extension of Apostol type Frobenius-Euler polynomials and numbers and investigate some basic identities and properties for these polynomials and numbers, including addition theorems, difference equations, derivative properties, recurrence relations and so on. Then, we provide integral representations, explicit formulas and relations for these polynomials and numbers. Moreover, we discover (p, q)-extensions of Carlitz's result [L. Carlitz, Mat. Mag., 32 (1959), 247–260] and Srivastava and Pintér addition theorems in [H. M. Srivastava, A. Pinter, Appl. Math. Lett., 17 (2004), 375–380].

The article is devoted to applications of 2-dimensional hyperbolic numbers and their algebraic 2ⁿ-dimensional extensions in modeling some genetic and cultural phenomena. Mathematical properties of hyperbolic numbers and their bisymmetric matrix representations are described in a connection with their application to analyze the following structures: alphabets of DNA nucleobases; inherited phyllotaxis phenomena; Punnett squares in Mendelian genetics; the psychophysical Weber-Fechner law; long literary Russian texts (in their special binary representations). New methods of algebraic analysis of the harmony of musical works are proposed, taking into account the innate predisposition of people to music. The hypothesis is put forward that sets of eigenvectors of matrix representations of basis units of 2ⁿ-dimensional hyperbolic numbers play an important role in transmitting biological information. A general hyperbolic rule regarding the oligomer cooperative organization of different genomes is described jointly with its quantum-information model. Besides, the hypothesis about some analog of the Weber-Fechner law for sequences of spikes in single nerve fibers is formulated. The proposed algebraic approach is connected with the theme of the grammar of biology and applications of bisymmetric doubly stochastic matrices. Applications of hyperbolic numbers reveal hidden interrelations between structures of different biological and physical phenomena. They lead to new approaches in mathematical modeling genetic phenomena and innate biological structures.

In this paper, we introduce the generalized dual hyperbolic Fibonacci numbers. As special cases, we deal with dual hyperbolic Fibonacci and dual hyperbolic Lucas numbers. We present Binet's formulas, generating functions and the summation formulas for these numbers. Moreover, we give Catalan's, Cassini's, d'Ocagne's, Gelin-Cesàro's, Melham's identities and present matrices related with these sequences.

Incorporating renewables in the power grid presents challenges for stability, reliability, and operational efficiency. Integrating Energy Storage Systems (ESS) offers a solution by managing unpredictable loads, enhancing reliability, and serving the grid. Hybrid storage solutions have gained attention for specific applications, suggesting higher performance in some respects. This paper compares the performance of hybrid energy storage systems (HESS) to single battery, evaluating their energy supply cost and environmental impact through optimization problems. The model generates an optimized dispatch, minimizing cost or environmental impact of supplying energy to a generic load while considering battery degradation. Seven technologies are assessed, with an example applied to an industrial site contrasting vanadium redox flow (VRFB) and lithium batteries considering the demand of a local load (building). The results also indicates that efficiency and degradation curves have the highest impact in the final costs and environmental functions on the various storage technologies assessed. For the simulations of the example case, a single system only outperforms the hybrid system in cases where lithium efficiency is higher than approximately 87% and vanadium is lower approximately 82%.

The existence of palindrome numbers of arbitrary length greater than 2 that are divisible by the prime number 37 is proven in this paper. The method to construct such a palindrome is also described. It turns out that a palindrome number divisible by 37 is also divisible by 3 and, consequently, divisible by 111.

Recently, a paper was written to reformulate and strengthen the theory of semi-structured complex numbers H (a new number set invented to enable division by zero). Whilst the paper had profound results, the application of this number set in many areas of engineering and science has not been fully explored. Consequently, the aim of this paper was to establish the number set H as a useful mathematical tool by providing an amalgam of results arising from its application in several areas of engineering and science. In the process, this paper makes four major contributions: these are (1) determining the product of the gradients of a horizontal and vertical line; (2) developing new Euler formulas relating exponential, trigonometric and hyperbolic functions in the 3D semi-structured complex number Euclidean space; (3) establishing semi-structured complex numbers as a better extension of the complex number set than quaternions; (4) resolving singularities that may arise in engineering and science equations (because of division by zero) to develop reasonable conclusions in the absence of experimental data. These results and their applications provide a firm foundation to advance the number set H as a useful mathematical tool.

Motivated by Kim-Kim [19] introduced the new type of degenerate poly- Bernoulli polynomials by means of the degenerate polylogarithm function. In this paper, we define the degenerate poly-Frobenius-Euler polynomials, called the new type of degenerate poly-Frobenius-Euler polynomials, by means of the degenerate polylogarithm function. Then, we derive explicit expressions and some identities of those numbers and polynomials.

Recently, Kim-Kim [13] have introduced polyexponential functions as an inverse to the polylogarithm functions, and constructed type 2 poly-Bernoulli polynomials. They have also introduced unipoly functions attached to each suitable arithmetic function as a universal concept. Inspired by their work, in this paper, we introduce a new class of the Frobenius-Genocchi polynomials. We derive the diverse formulas and identities covering some summation formulas, derivative formula and correlations with Bernoulli polynomials and numbers, Stirling numbers of the both kinds, degenerate Frobenius-Genocchi polynomials and degenerate Frobenius-Euler polynomials. Moreover, by using the unipoly function as following Kim-Kim's work in <cite>Kim1</cite>, we consider degenerate unipoly-Frobenius-Genocchi polynomials and investigate some formulas and relationships with Daehee numbers, degenerate Frobenius-Genocchi numbers and Stirling numbers of the first kind. Finally, we obtain an Gaussian integral representation of the Frobenius-Genocchi polynomials in terms of the 2-variable Hermite polynomials.

In the paper, the authors establish two identities, which can be regarded as nonlinear differential equations, for the generating function of Eulerian polynomials, find two identities for the Stirling numbers of the second kind, and present two identities for Eulerian polynomials and higher order Eulerian polynomials, pose two open problems about summability of two finite sums involving the Stirling numbers of the second kind. Some of these conclusions meaningfully and significantly simplify several known results.

In the paper, the authors find two explicit formulas and recover a recursive formula for the generalized Motzkin numbers. Consequently, the authors deduce two explicit formulas and a recursive formula for the Motzkin numbers, the Catalan numbers, and the restricted hexagonal numbers respectively.

In recent years, the abundance of streaming data has posed significant challenges for traditional machine learning algorithms due to the dynamic nature and high volume of data. To address these challenges, this paper presents an adaptive supervised learning framework designed for data streams, leveraging the power of reproducing kernel Hilbert spaces (RKHS) while incorporating a data sparsity constraint. The proposed framework aims to adaptively update the model to efficiently handle the continuous arrival of streaming data while ensuring accurate and reliable predictions. The primary objective is to exploit the RKHS framework, which provides a rich mathematical structure for learning in high-dimensional feature spaces. By utilizing RKHS, the model can capture complex patterns and nonlinear relationships in the streaming data. Furthermore, the framework incorporates a data sparsity constraint to address the issue of limited resources and computational efficiency. The constraint promotes the selection of a subset of relevant features, reducing the dimensionality of the problem and enhancing the scalability of the learning algorithm. This constraint not only improves computational efficiency but also mitigates the effects of noisy or irrelevant features, leading to more robust and accurate predictions. To achieve adaptability, the proposed framework employs an online learning approach that incrementally updates the model parameters as new data arrives. This allows the model to adapt to concept drift and changing data distributions, ensuring its relevance and effectiveness over time. The adaptation process is guided by a mechanism that balances the exploitation of current knowledge with the exploration of new information, enabling the model to gradually evolve and refine its predictions. Experimental evaluations on various benchmark datasets demonstrate the efficacy of the proposed framework in handling streaming data. The results indicate that the adaptive supervised learning approach in RKHS with a data sparsity constraint outperforms traditional batch learning methods and exhibits superior accuracy, scalability, and adaptability in dynamic data stream scenarios. Overall, this research contributes to the development of adaptive supervised learning methods for data streams, highlighting the effectiveness of RKHS and the significance of incorporating a data sparsity constraint. The proposed framework offers a promising solution to handle the challenges posed by streaming data, paving the way for real-time, efficient, and accurate learning in diverse application domains.

Filter Media (FM) sourced from recycled organic and mineral material offers a low cost and effective means of treating urban stormwater. Using recycled materials rather than from an increasingly scarce source of virgin materials (typically sandy loam soil) can ensure a sustainable long-term economy and environment. This paper presents results from the laboratory analysis and mathematical modeling to highlight the performance of recycled organic and mineral material in removing nutrients and metals from stormwater. Analysis included physical and chemical characterisation such as particle size distribution, saturated hydraulic conductivity (Ksat), bulk density, effective cation exchange capacity, and pollutant removal performance. Design mixes (DM), comprising a combination of organic and mineral materials, were characterised and used to develop/derive modelling design within the Model for Urban Stormwater Improvement Conceptualisation (MUSIC v6) [1]. Comparison is made to the Adoption Guidelines for Stormwater Biofiltration Systems - Summary Report [2] which were based on the Facility for Advancing Water Biofiltration (FAWB) guidelines to assist in the development of biofiltration systems, including the planning, design, construction and operation of those systems. An observed outcome from over two decades of biofiltration guideline development has been the exclusion of alternative biofilter materials due to claims of excessive leaching. Results from this study indicate that high nutrient and metal removal rates can be achieved over a range of hydraulic conductivities using design mixes of recycled organic and mineral materials that have a demonstrated equivalence to existing guideline specifications.

In this study, a critical review was carried out using the Web of ScienceTM Core Collection database to analyse the scientific literature published to date to identify lines of research and future perspectives on the presence of chemical pollutants in beer brewing. Beer is one of the world's most popular drinks and the most consumed alcoholic beverage. However, a widespread challenge with potential implications for human and animal health is the presence of physical, chemical and/or microbiological contaminants in beer. Biogenic amines, heavy metals, mycotoxins, nitrosamines, pesticides, acrylamide, phthalates, bisphenols, microplastics and, to a lesser extent, hydrocarbons (aliphatic chlorinated and polycyclic aromatic), carbonyls, furan-derivatives, polychlorinated biphenyls and trihalomethanes are the main chemical pollutants found during the beer brewing process. Pollution sources include raw materials, technological process steps, the brewery environment and packaging materials. Different chemical pollutants have been found during the beer brewing process, from barley to beer. Brewing steps such as steeping, kilning, mashing, boiling, fermentation and clarification are critical in reducing the levels of many of these pollutants. As a result, their residual levels are usually below the maximum levels allowed by international regulations.

In this research first, a sequence of properties called delta is assigned to each prime number and then examined. Deltas are only dependent on the distribution of prime numbers, so the results obtained for the delta distribution can be considered as a proxy for the distribution of prime numbers. The first observation was that these properties are not unique and different prime numbers may have the same value of delta of a given order. It was found that a small number of deltas cover a large portion of prime numbers, so by recognizing repetitive deltas, the next prime numbers can be predicted with a certain probability, but the most important observation of this study is the normal distribution of deltas. This research has not tried to justify the obtained observations and instead of answering the questions, it seeks to ask the right question.

Creative thinking is increasingly recognised as an essential ability that should be part of school curricula. Given the move towards online learning and assessment, we investigate whether mathematical creativity can be assessed at-scale in the Numbers game, an arithmetic game in Math Garden, a popular online math practice platform. In the Numbers game, a generalisation of the 24 Game, children are asked to figure out how to compute a target number using basic arithmetic operations and a given set of numbers. We argue that creative thinking is required when the search space is complex, and propose that the base-pattern, i.e., the sequence of the operations needed to solve a Numbers game item, indicates search space complexity. We then demonstrate that items with disordered base-patterns are more likely to require mathematical creativity to figure out. Specifically, our analysis shows that for items with only one solution sequence, those with disordered base-patterns are more difficult and take longer to solve compared to items with ordered base-patterns. For items where multiple solution sequences are possible, nine times out of ten children choose ordered over disordered base-patterns. We conclude that the Numbers game has potential for assessing mathematical creativity at-scale.

In this paper, we define new kind of Daehee numbers, the degenerate Daehee numbers of the third kind, using degenerate log function as generating function. We obtain some identities for the degenerate Daehee numbers of the third kind associated with the Daehee, degenerate Daehee and degenerate Daehee numbers of the second kind. In addition, we derive a differential equation associated with the degenerate log function. We deduce some identities from the differential equation.

In the paper, the authors express the Fuss–Catalan numbers as several forms in terms of the Catalan–Qi function, find some analytic properties, including the monotonicity, logarithmic convexity, complete monotonicity, and minimality, of the Fuss–Catalan numbers, and derive a double inequality for bounding the Fuss–Catalan numbers.

Carnivorous waterbirds have been reported to enhance wetlands diversities, control pests, bio-indicators of ecological conditions, and act as indicators of potential disease outbreaks. They also provide important ecosystem services to indigenous communities residing along wetlands. Regardless of their importance, global waterbirds populations are declining. Despite their ecological contribution to wetlands functions and ecosystem services, piscivore waterbirds have been re-ported to compete with piscivorous fish and fishermen. The aim of this study was to compare piscivorous waterbirds species and functional diversity between fish protected areas (FPA) and non-fish protected area (NFPA) in two tributaries of Zambezi River in Namibia. We also measured the degree to which patterns of guild structure between piscivore waterbirds and fish were similar between FPA and non-FPA. Species diversities were determined by calculating the Shannon Diversity Index. We calculated taxonomic diversity, functional diversity and community-based trait diversity indices for each tributary. Generally, species diversities were not significantly different between FPA and NFPA (P > 0.05). However, all functional diversities indices calculated were significantly different between FPA an NFPA (P < 0.05 for all comparisons). This was partly due to food item, weight and movement type that were associated with each site.

Traumatic cardiac arrest has a high mortality. We report the case of a 65 years old women who presented a cardiac arrest following traffic accident. Examinations concluded to a tension gastro thorax with a delayed diagnosis. Naso gastric tube insertion was for many authors a temporary care while waiting for surgery. We report on the diagnostic difficulties found in the literature and a recent review of published cases.

Mass customization, small batch sizes, high variability of product types and a changing product portfolio during the life cycle of an industrial plant are current trends in the industry. Due to an increasing decoupling of the development of software and hardware components in an industrial context, compatibility problems within industrial control systems arise more and more frequently. In this publication, a strategy concept for compatibility testing is derived and discussed by means of literature review and applied research. This 4-phased strategy concept identifies incompatibilities between software and hardware components in the industrial control environment and enables test engineers to detect problems at an early stage. By automating the compatibility test on an external I-PC, the test can be run both when new software is installed on the industrial controller and when the controller is restarted. Thus, changes to the components are constantly detected and incompatibilities are avoided. Furthermore, early incompatibility detection can ensure that a system remains permanently operational. Based on a discussion, additionally strategies are identified to consolidate the robustness and applicability of the presented concept.

In this work we explore the utilization of plasmonic resonance (PR) in silver nanowires to enhance the performance of organic solar cells. Plasmonic resonance is a phenomenon in which nanoscale conductive materials exhibit oscillation of conduction electrons, resulting in the creation of an electric field. Enhancing light absorption is crucial for improving organic solar cell efficiency and incorporating metallic nanostructures to induce surface plasmon resonance (SPR) shows promise in achieving this goal. We discuss the two key mechanisms of plasmonic effects: far-field scattering and near-field resonance modes. Far-field scattering extends the optical path of incident light, while near-field plasmonic effects involve localized surface plasmon resonance (LSPR) and plasmonic cavity modes, enhancing absorption by strengthening electric fields near the nanostructures. Silver nanowires are the focus of this study, and finite-difference time-domain (FDTD) simulation software is used to investigate their plasmonic resonance behavior in a ZnO/Silver nanowires/ZnO (ZAZ) electrode structure. The simulations reveal the dominance of LSPR in this configuration, with intense electric fields inside the nanowire and propagation into the surrounding medium, offering opportunities for enhanced light absorption in the organic solar cell's active layer.

We demonstrate a new quantitative method to the sieve of Eratosthenes, which is an alternative to the sieve of Legendre. In this method, every element of a given set is sifted out once only, and therefore, this method is free of the Mobius function and of the parity barrier. Using this method, we prove that every sufficiently large even number is the sum of two primes, and that every even number is the difference of two primes in infinitely many ways.

In this paper, we introduce a new type of degenerate poly-Fubini polynomials and numbers, are called degenerate poly-Fubini polynomials and numbers, by using the degenerate polylogarithm function and derive several properties on the degenerate poly-Fubini polynomials and numbers. In the last section, we also consider the degenerate unipoly-Fubini polynomials attached to an arithmetic function, by using the degenerate polylogarithm function and investigate some identities of those polynomials. In particular, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.

In this paper, we construct the degenerate poly-Fubini polynomials, called the type 2 degenerate poly-Fubini polynomials, by using the modified degenerate polyexponential function and derive several properties on the degenerate poly-Fubini polynomials and numbers. In the last section, we introduce type 2 degenerate unipoly- Fubini polynomials attached to an arithmetic function, by using the modified degenerate polyexponential function and investigate some identities for those polynomials. Furthermore, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.

In this paper, we introduce a new class of Laguerre-based Hermite-Fubini numbers and polynomials and investigate some properties of these polynomials. We establish summation formulas of these polynomials by summation techniques series. Furthermore, we derive symmetric identities of Laguerre-based Hermite-Fubini numbers and polynomials by using generating functions.

Recent technological advancements such as IoT and Big Data have granted industries extensive access to data, opening up new opportunities for integrating Artificial Intelligence (AI) across various applications to enhance production processes. We can cite two critical areas where AI can play a key role in industry such as product quality control and predictive maintenance. This paper presents a survey of AI applications in the domain of Industry 4.0, with a specific focus on product quality control and predictive maintenance. Experiments were conducted using two datasets, incorporating different machine learning and deep learning models from the literature. Furthermore, this paper provides an overview of the AI solution development approach for product quality control and predictive maintenance. This approach includes several key steps, such as data collection, data analysis, model development, model explanation, and model deployment.

Following the development of tyrosine kinase inhibitors (TKI), the survival of patients with chronic myeloid leukaemia (CML) drastically improved. With the introduction of these agents, CML is now considered a chronic disease, for some patients. Taking into consideration the side effects, toxicity, and high cost, discontinuing TKIs became a goal for patients with chronic phase CML. Patients who achieved deep molecular response (DMR) and discontinued TKI, remained in treatment-free remission (TFR). Currently, the data from the published literature demonstrate that 40-60% of patients achieve TFR, with relapses occurring within the first six months. In addition, almost all patients who relapsed regained a molecular response upon re-treatment, indicating TKI discontinuation is safe. However, there is still a gap in the understanding the mechanisms behind TFR, and whether there are prognostic factors that can predict the best candidates who qualify for TKI discontinuation with a view to keeping them in TFR. Furthermore, the information about a second TFR attempt and the role of gradual de-escalation of TKI before complete cessation is limited. This review highlights the factors predicting success or failure of TFR. In addition, it ex-amines the feasibility of a second TFR attempt after the failure of the first one, and the current guidelines concerning TFR in clinical practice.

This work is divided into two parts. In the first one the combinatorics of a new class of randomly generated objects, exhibiting the same properties as the distribution of prime numbers, is solved and the probability distribution of the combinatorial counterpart of the n-th prime number is derived, together with an estimate of the prime-counting function π(x). A proposition equivalent to the Prime Number Theorem (PNT) is proved to hold, while the equivalent of the Riemann Hypothesis (RH) is proved to be false with probability 1 (w.p. 1) for this model. Many identities involving Stirling numbers of the second kind and harmonic numbers are found, some of which appear to be new. The second part is dedicated to generalizing the model to investigate the conditions enabling both PNT and RH. A model representing a general class of random integer sequences is found, for which RH holds w.p. 1. The prediction of the number of consecutive prime pairs, as a function of the gap d, is derived from this class of models and the results are in agreement with empirical data for large gaps. A heuristic version of the model, directly related to the sequence of primes, is discussed and new integral lower and upper bounds of π(x) are found.

Within the framework of multiblock data analysis, a unified approach of supervised methods is discussed. It encompasses multiblock redundancy analysis (MB-RA) and multiblock partial least squares (MB-PLS) regression. Moreover, we develop new supervised strategies of multiblock data analysis, which can be seen as variants of one or the other of these two methods. They are respectively refered to as multiblock weighted redundancy analysis (MB-WRA) and multiblock weighted covariate analysis (MB-WCov). The four methods are based on the determination of latent variables associated with the various blocks of variables. They are derived from clear optimization criteria whose aim is to maximize either the sum of the covariances or the sum of squared covariances between the latent variable associated with the response block of variables and the block latent variables associated with the various explanatory blocks of variables. We also propose indices to help better interpreting the outcomes of the analyses. The methods are illustrated and compared based on simulated and real datasets.

The normal and Poisson distribution assumptions in the normal-Poisson mixed effects regression model are often too restrictive for many real count data. Several works have independently relaxed the Poisson conditional distribution assumption for counts or the normal distribution assumption for random effects. This work couples some recent advances in these two regards to develop a skew t–discrete gamma regression model in which the count outcomes have full dispersion flexibility and random effets can be skewed and heavy tailed. Inference in the model is achieved by maximum likelihood using pseudo-adaptive Gaussian quadature. The use of the proposal is demonstrated on a popular owl sibling negotiation data. It appears that, for this example, the proposed approach outperforms models based on normal random effects and the Poisson or negative binomial count distribution.

Most existing flexible count regression models allow only approximate inference. Balanced discretization is a simple method to produce a mean-parametrizable flexible count distribution starting from a continuous probability distribution. This makes easy the definition of flexible count regression models allowing exact inference under various types of dispersion (equi-, under- and overdispersion). This study describes maximum likelihood (ML) estimation and inference in count regression based on balanced discrete gamma (BDG) distribution and introduces a likelihood ratio based latent equidispersion (LE) test to identify the parsimonious dispersion model for a particular dataset. A series of Monte Carlo experiments were carried out to assess the performance of ML estimates and the LE test in the BDG regression model, as compared to the popular Conway-Maxwell-Poisson model (CMP). The results show that the two evaluated models recover population effects even under misspecification of dispersion related covariates, with coverage rates of asymptotic 95% confidence interval approaching the nominal level as the sample size increases. The BDG regression approach, nevertheless, outperforms CMP regression in very small samples (n = 15 − 30), mostly in overdispersed data. The LE test proves appropriate to detect latent equidispersion, with rejection rates converging to the nominal level as the sample size increases. Two applications on real data are given to illustrate the use of the proposed approach to count regression analysis.

In this paper, we investigate p-adic q-integral (q-Volkenborn integral) on Z_p of p-adic gamma function via their Mahler expansions. We also derived two q-Volkenborn integrals of p-adic gamma function in terms of q-Daehee polynomials and numbers and q-Daehee polynomials and numbers of the second kind. Moreover, we discover q-Volkenborn integral of the derivative of p-adic gamma function. We acquire the relationship between the p-adic gamma function and Stirling numbers of the first kind. We finally develop a novel and interesting representation for the p-adic Euler constant by means of the q-Daehee polynomials and numbers.

In this paper we investigate some properties of square-free numbers. In particular, we study a function that counts the number of square-free numbers no larger than the given number n. We show that this function has asymptotics consistent with the predictions of the Riemann hypothesis and use this to obtain new and better estimates of the density of square-free numbers. Finally, we present a well-known generalization of the concept of a square-free number to monoids and research on square-free factorizations.

Motivation by the definition of the type 2 poly-Bernoulli polynomials introduced by Kim-Kim [9], in the present paper, we consider a new class of new generating function for the Fubini polynomials, called the type 2 poly-Fubini polynomials by means of the polyexponential function. Then, we derive some useful relations and properties. We show that the type 2 poly-Fubini polynomials equal a linear combination of the classical of the Fubini polynomials and Stirling numbers of the first kind. In a special case, we give a relation between the type 2 poly-Fubini polynomials and Bernoulli polynomials of order r. Moreover, inspired by the definition of the unipoly-Bernoulli polynomials introduced by Kim-Kim [9]. We introduce the type 2 unipoly-Fubini polynomials by means of unipoly function and give multifarious properties including derivative and integral properties. Furthermore, we provide a correlation between the unipoly-Fubini polynomials and the classical Fubini polynomials.

In this paper, we consider a new class of polynomials which is called the multi-poly-Euler polynomials. Then, we investigate their some properties and relations. We provide that the type 2 degenerate multi-poly-Euler polynomials equals a linear combination of the degenerate Euler polynomials of higher order and the degenerate Stirling numbers of the first kind. Moreover, we provide an addition formula and a derivative formula. Furthermore, in a special case, we acquire a correlation between the type 2 degenerate multi-poly-Euler polynomials and degenerate Whitney numbers.

Inspired by the definition of degenerate multi-poly-Genocchi polynomials given by using the degenerate multi-polyexponential functions. In this paper, we consider a class of new generating function for the degenerate multi-poly-Bernoulli polynomials, called the type 2 degenerate multi-poly-Bernoulli polynomials by means of the degenerate multiple polyexponential functions. Then, we investigate their some properties and relations. We show that the type 2 degenerate multi-poly-Bernoulli polynomials equals a linear combination of the weighted degenerate Bernoulli polynomials and Stirling numbers of the first kind. Moreover, we provide an addition formula and a derivative formula. Furthermore, in a special case, we acquire a correlation between the type 2 degenerate multi-poly-Bernoulli numbers and degenerate Whitney numbers.

Mathematical modelling of bioprocesses has a long and notable history, with eminent contributions from fields including microbiology, ecology, biophysics, chemistry, statistics, control theory and mathematical theory. This richness of ideas and breadth of concepts provide great motivation for inquisitive engineers and intrepid scientists to try their hand at modelling, and this collaboration of disciplines has also delivered significant milestones in the quality and application of models for both theoretical and practical interrogation of engineered biological systems. The focus of this review is the anaerobic digestion process, which, as a technology that has come in and out of fashion, still remains a fundamental process for addressing the global climate emergency. Whether with conventional anaerobic digestion systems, biorefineries, or other anaerobic technologies, mathematical models are important tools that are used to design, monitor, control and optimise the process. Both highly structured, mechanistic models and data-driven approaches have been used extensively over half a decade, but recent advances in computational capacity, scientific understanding and diversity and quality of process data, presents an opportunity for the development of new modelling paradigms, augmentation of existing methods, or even incorporation of tools from other disciplines, to ensure that anaerobic digestion research can remain resilient and relevant in the face of emerging and future challenges.

In this paper, we investigate p-adic q-integral (q-Volkenborn integral) on ℤ_{p} of p-adic gamma function via their Mahler expansions. We also derived two q-Volkenborn integrals of p-adic gamma function in terms of q-Daehee polynomials and numbers and q-Daehee polynomials and numbers of the second kind. Moreover, we discover q-Volkenborn integral of the derivative of p-adic gamma function. We acquire the relationship between the p-adic gamma function and Stirling numbers of the first kind. We finally develop a novel and interesting representation for the p-adic Euler constant by means of the q-Daehee polynomials and numbers.

In this paper, we introduce a new class of $(p,q)$-analogue type of Fubini numbers and polynomials and investigate some properties of these polynomials. We establish summation formulas of these polynomials by summation techniques series. Furthermore, we consider some relationships for $(p,q)$-Fubini polynomials associated with $(p,q)$-Bernoulli polynomials, $(p,q)$-Euler polynomials and $(p,q)$-Genocchi polynomials and $(p,q)$-Stirling numbers of the second kind.

In this paper, we investigate the new degenerate Daehee polynomials and numbers which are called the degenerate Daehee polynomials of the second kind, and derive some new and interesting identities and properties of those polynomials.

The main goal of the present paper is to construct some families of the Carlitz’s q-Bernoulli polynomials and numbers. We firstly introduce the modified Carlitz’s q-Bernoulli polynomials and numbers with weight (α, β) and investigate their some explicit properties and identities arising from the bosonic q-Volkenborn integral on ℤ_p. We then define the modified degenerate Carlitz’s q-Bernoulli polynomials and numbers with weight (α, β) and obtain some recurrence relations and other identities. Moreover, we derive some correlations with the modified Carlitz’s q-Bernoulli polynomials with weight (α, β), the modified degenerate Carlitz’s q-Bernoulli polynomials with weight (α, β), the Stirling numbers of the first kind and second kind.

In the paper, by the Faά di Bruno formula, several identities for the Bell polynomials of the second kind, and an inversion theorem, the authors simplify coefficients of two families of nonlinear ordinary differential equations for the generating function of the Catalan numbers and discover inverses of fifteen closely related lower triangular integer matrices.

In fuzzy decision problems, the ordering of fuzzy numbers is the basic problem. Among which, the fuzzy preference relation is the reasonable one to represent preference relation by a fuzzy membership function. This paper studies the Nakamura’s and Kołodziejczyk’s preference relations. Eight cases of representing different level of overlapping between two triangular fuzzy numbers are considered. We analyze the ranking behaviors of all possible combinations of decomposition and intersection of two fuzzy numbers for the Nakamura’s and Kołodziejczyk’s preference relations of these test cases. The results indicate that the decomposition and intersection can affect the fuzzy preference relations, thereby the final total order relation of fuzzy numbers.

In this article, the usual factorials and binomial coefficients have been generalized and extended to the negative integers. Basing on this generalization and extension, a new kind of polynomials has been proposed, which led directly to the non-classical hypergeometric orthogonal polynomials and the non-classical second-order hypergeometric linear DEs. The resulting polynomials can be used in non-relativistic and relativistic QM, particularly, in the case of the Schrödinger equation, and Dirac equations for an electron in a Coulomb potential field.

Understanding the relationships between land cover/urban structure patterns and air pollutants is key to sustainable urban planning and development. In this study, we employ a mobile monitoring method to collect PM_2.5 and BC data in the city of Philadelphia, PA during the summer of 2019 and apply the Structure of Urban Landscapes (STURLA) methodology to examine relationships between urban structure and atmospheric pollution. We find that, while PM_2.5 and BC vary by STURLA class, many of the differences in pollutant concentrations between classes are not significant. However, we also find that the proportions in which STURLA components are present throughout the urban landscape can be used to predict urban air pollution. Among frequently sampled STURLA classes, gpl hosted the highest PM_2.5 concentrations on average (16.60 _± 4.29 µg/m³), while tgbwp hosted the highest BC concentrations (2.31 _± 1.94 µg/m³). Furthermore, STURLA combined with machine learning modeling was able to correlate PM_2.5 (R²= 0.68, RMSE 2.82 µg/m³) and BC (R² = 0.64, RMSE 0.75 µg/m³) concentrations with the urban landscape and spatially interpolate concentrations where sampling did not take place. These results demonstrate the efficacy of the STURLA methodology in modeling relationships between air pollution and land cover/urban structure patterns.

In this article, we discuss the concept of infinity (∞), infinitesimals â–¡ , and a new mathematical operation that expands calculations into the realm of the infinite. This breakthrough allows for the computation of infinite numbers, leading to new possibilities for mathematical research and practical applications. By introducing infinite numbers, numerous advantages arise, such as accurately representing limits, divergences in integrals and infinite series, and extending calculations to infinity. This mathematical advancement opens doors to exploring uncharted territories, offering valuable insights and breakthroughs.

We developed some questions regarding twin prime conjecture by examining the bounded gaps between composite numbers and explained the relations between the formulas regarding composite numbers. Then we proved the twin prime conjecture to answer the new questions raised by proving the expression: \[ \liminf_{n\to\infty} (c_n_+_1 - c_n) \ge 10. \] In addition, we prove many different approaches have been and it has been proved that there must be a twin prime in the range $(p_k_2\cdot p_k_1, p^2_k_1)$, where $p \neq 2$.

This short technical note proves some observed novel corollaries of the Collatz Conjecture. For this, a unique ‘Collatz Representation’ of Hailstone numbers has been employed. After that, an extension of this corollary is explained using the fact that only certain differences in the last digit of the ‘Collatz Representation’ gives a whole number difference between numbers. A more general result is obtained. The final proof is for a corollary that some numbers the same number of steps to reach 1 if the Collatz Conjecture is true and all variables are whole numbers.

Starting with a positive odd integer $n$, apply a function $f(n)$ repetitively such that $f(n) = 3n + 1$ if $n \not\equiv 0 (\text{mod}\:2)$ or $f(n) = n/2$ if $n \equiv 0 (\text{mod}\:2)$. Let the sequence of all the odd integers obtained form an orbit $n, f^1(n) , f^2(n), \ldots, f^k(n), f^{k + 1}(n)$. The $k^{th}$ odd integer is written as $f^k(n) = (3^k + \mathbf{B(k)})/2^{\mathbf{z(k)}}$ where $\mathbf{B(k)}$ is the collection of terms without the coefficient $n$. We show that when $1 - \mathbf{B(k)}/(2^{\mathbf{z(k)}} n) \approx 1$, the condition for unbounded orbit is given by OEIS A022921. Further, if there exists at least one orbit for which $f^{k + 1}(n) = n$ when $(k + 1) < 3$, we show that there exists no orbit with $k + 1 > 3$ for which $f^{k + 1}(n) = n$. The final result is that the odd integers that can violate the Collatz conjecture are smaller than 5.