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.

The growing need to use renewable sources and the current difficulty in spreading the electricity grid in a widespread manner raise the question of how to respond to the need for more electricity immediately. The idea behind this study is to power a horizontal axis wind turbine with the air flow generated for cooling a stationary internal combustion engine. The power extracted from this solution is significantly lower than that of the internal combustion engine (about 0.3%) and could be advantageous only in limited contexts. Installation costs are limited because many elements deriving from wind variability can be removed or simplified.

The realization of the third-generation artificial intelligence (AI) requires the evolution from perceptual intelligence to cognitive intelligence, where knowledge graphs may not meet the practical needs anymore. Based on the dual channel theory, cognitive graphs are established and developed through coordinating the implicit extraction module and the explicit reasoning module as well as integrating knowledge graphs, cognitive reasoning and logical expressions, which have achieved successes in multi-hop question answering. It is desired for cognitive graphs to be widely used in advanced AI applications such as large-scale knowledge representations and intelligent responses, promoting the development of Al dramatically. This review discusses cognitive graphs systematically and elaborately, including basic concepts, generations, theories and technologies. Moreover, we try to predict the development of cognitive intelligence in the short-term future and further enlighten more researches and studies.

The paper describes a method of solving some stochastic processes using generating functions. A general theorem of generating functions of a particular type is derived. A generating function of this type is applied to a stochastic process yielding polynomial time algorithms for certain partitions. The method is generalized to a stochastic process describing a rather general linear transform. Finally, the main idea of the method is used in deriving a theoretical polynomial time algorithm to the knapsack problem.

In this paper, we introduce a new generalization of the Hermite polynomials via (p, q)-exponential generating function and investigate several properties and relations for mentioned polynomials including derivative property, explicit formula, recurrence relation, integral representation. We also de…ne a (p, q)-analogue of the Bernstein polynomials and acquire their some formulas. We then provide some (p, q)-hyperbolic representations of the (p, q)-Bernstein polynomials. In addition, we obtain a correlation between (p, q)-Hermite polynomials and (p, q)-Bernstein polynomials.

We study various properties of the polygonal numbers; such as their recurrence relations; fundamental identities; weighted binomial and ordinary sums; partial sums and generating functions of their powers; and a continued fraction representation for them. A feature of our results is that they are presented naturally in terms of the polygonal numbers themselves and not in terms of arbitrary integers; unlike what obtains in most literature.

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, 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.

The main aim of this paper is to investigate multifarious properties and relations for the gamma distribution. The approach to reach this purpose will be introducing a special polynomial including gamma distribution. Several formulas covering addition formula, derivative property, integral representation and explicit formula are derived by means of the series manipulation method. Furthermore, two correlations including Bernoulli and Euler polynomials for gamma distribution polynomials are provided by utilizing of their generating functions.

The present paper discusses a study of a class of Charlier matrix polynomials and its generalized analogue. Certain generating matrix functions, recurrence matrix relations, matrix differential equation, summation formulas and many new results have been discussed for these matrix polynomials. Weisner's group theoretic method is used to obtain matrix generating relations for Charlier matrix polynomials and the details of this method were given in this paper. Finally, we will discuss only briefly the procedure followed.

Flood routing in flood forecasting issue, calculation the height of flood bands, determining the river boundaries, and estimation of protective facilities for flood –exposed building is applicable. In many cases, due to the lack of measuring stations, the status of the upstream flood generating hydrograph is not known. The purpose of this study is to present an integrated method comprising of an optimization model and a hydrodynamic numerical model for flood modeling to determine the upstream hydrograph using the provided hydrograph at the downstream measuring station of a river. The routing procedure consists of three steps: (1) generating a hypothetical upstream hydrograph using genetic algorithm method; (2) hydrodynamic modeling using a numerical simulation model for flood routing according to the hypothetical hydrograph which is generated in the first step; (3) compare the calculated and observed hydrograph in downstream by using a fitness function. This recommended procedure was named as Reverse Flood Routing Method (RFRM) and was then applied to Karun River, the largest river in Iran. Comparing the generated upstream hydrograph by the RFRM model with the corresponding measured hydrograph at Ahvaz hydrometric station, as an ungauged river location, shows the high accuracy of the recommended model in this study.

This paper studies the goodness of fit test for the bivariate Hermite distribution. Specifically, we propose and study a Cramér-von Mises-type test based on the empirical probability generation function. The bootstrap can be used to consistently estimate the null distribution of the test statistics. A simulation study investigates the goodness of the bootstrap approach for finite sample sizes.

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 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.

In this article, the Legendre-Gould Hopper polynomials are combined with Sheffer sequences to introduce certain mixed type special polynomials. Generating functions, differential equations and certain other properties of Legendre-Gould Hopper based Sheffer polynomials are derived. Further, operational and integral representations providing connections between these polynomials and known special polynomials are established. Certain identities and results for some members of these new mixed polynomials are also obtained. Finally, the determinantal definitions of Legendre-Gould Hopper based Sheffer polynomials are also given.

Environmental assessment is a concept that has been designed to facilitate the present generation to meet their needs without compromising the ability of future generations to meet their own needs as well. Thus, this concept has drawn significant attention from various scholars, researchers and industrial practitioners around the world over the past three decades. Life Cycle Environmental Assessment (LCEA) is a widely metric used to assess the potential ecological impacts, which can be caused by electricity generating supply systems or by other systems than power production plants. However, the current LCEA model is biased and ineffective. Because, its omits factors that are increasingly contributing to the ecological degradation. This study has identified the omitted factors through a critical analysis of a set of previous journal articles conducted in the energy sector. In light of this, this study has developed a novel LCEA framework addressing those blind spots. The framework developed in this study is holistic in nature including all the life cycle stages of a power supply system such as Extraction of the Raw Material (ERM), Transport of Raw Material (TRM), Conversion of Raw into Electricity (CRE), and Transmission and Distribution of Electricity (TDE) to the end users. The novel developed LCEA model has been tested and applied to nine power generation plants such as coal, gas, nuclear, biomass, geothermal, hydro, solar thermal, wind onshore and wind offshore. The results have demonstrated that of conventional technologies including coal, gas, and nuclear, coal energy generating source has got the highest life cycle greenhouse gas Grid Emission Factor (GEF) of 2866 kg CO2e/MWh, followed by gas with 728 kg CO2e/MWh, and nuclear has got the least GEF of 35 kg CO2e/MWh. Whereas of renewable energy sources biomass has got the highest GEF of 1508 kg CO2e/MWh, followed by solar thermal with 46.6 kg CO2e/MWh, hydro 39 kg CO2e/MWh, wind offshore 25.25 kg CO2e/MWh, wind onshore 10.1 kg CO2e/MWh, and geothermal closes the ranking with 6.23 kg CO2e/MWh.

It is quite easy to stochastically distort an original count variable to obtain a new count variable with relatively more variability than in the original variable. Many popular overdispersion models (variance greater than mean) can indeed be obtained by mixtures, compounding or randomlystopped sums. There is no analogous stochastic mechanism for the construction of underdispersed count variables (variance less than mean), starting from an original count distribution of interest. This work proposes a generic method to stochastically distort an original count variable to obtain a new count variable with relatively less variability than in the original variable. The proposed mechanism, termed condensation, attracts probability masses from the quantiles in the tails of the original distribution and redirect them toward quantiles around the expected value. If the original distribution can be simulated, then the simulation of variates from a condensed distribution is straightforward. Moreover, condensed distributions have a simple mean-parametrization, a characteristic useful in a count regression context. An application to the negative binomial distribution resulted in a distribution allowing under, equi and overdispersion. In addition to graphical insights, fields of applications of special cases of condensed Poisson and condensed negative binomial distributions were pointed out as an indication of the potential of condensation for a flexible analysis of count data

In this paper, we introduce a new generalization of the r-Stirling numbers of the second kind based on the q-numbers via an exponential generating function. We investigate their some properties and derive several relations among q-Bernoulli numbers and polynomials, and newly de…ned (q, r, w)-Stirling numbers of the second kind. We also obtain q-Bernstein polynomials as a linear combination of (q, r, w)-Stirling numbers of the second kind and q-Bernoulli polynomials in w.

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].

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 the paper, the authors consider the generating functions of the Hermite polynomials and their squares, present explicit formulas for higher order derivatives of the generating functions of the Hermite polynomials and their squares, which can be viewed as ordinary differential equations or derivative polynomials, find differential equations that the generating functions of the Hermite polynomials and their squares satisfy, and derive explicit formulas and recurrence relations for the Hermite polynomials and their squares.

This article aims to identify the generating function of modi…ed Apostol type q-Bernoulli polynomials. With the aid of this generating function, some properties of modi…ed Apostol type q-Bernoulli polynomials are given. It is shown that aforementioned polynomials are q-Appell. Hence, we make use of these polynomials to have applications on q-Umbral calculus. From those applications, we derive some theorems in order to get Apostol type modi…ed q-Bernoulli polynomials as a linear combination of some known polynomials which we stated in the paper.

In the Brazilian Amazon, rural settlements are increasingly isolated by large-scale production farms, jeopardizing their sustainability and the good living of family farmers. Works were carried out in settlements to measure sustainability. However, the majority does not consider the participation and the collectively of those involved. In this way, we propose to evaluate, in a collective and participatory way, the sustainability and good living of the SDP São Paulo Rural Settlement, of the northern Amazon of Mato Grosso. We used the didactic-pedagogical method Circle of Sustainability, developed from five points: 1st - circle of investigation of generating themes; 2nd - circle of the history of the subject world; 3rd - circle of diagnosis of rural settlements; 4th - circle of exchange of knowledge; and 5th - circle of sustainable perceptions and narratives. The historical, socioeconomic and cultural characterization of the settlement allowed us to understand how sustainability and good living are being built in the settlement history process. Sustainability and good living are dialectical processes, are under construction, in movement.

Recently, different extensions of the fractional derivative operator are found in many research papers. The main aim of this paper is to establish an extension of the extended Caputo fractional derivative operator. The extension of an extended fractional derivative of some elementary functions derives by considering an extension of beta function which includes the Mittag-Leffler function in the kernel. Further, an extended fractional derivative of some familiar special functions, the Mellin transforms of newly defined Caputo fractional derivative operator and the generating relations for extension of extended hypergeometric functions also presented in this study.

The main objective of this present paper is to introduce further extension of extended Caputo fractional derivative operator and establish the extension of an extended fractional derivative of some known elementary functions. Also, we investigate the extended fractional derivative of some familiar special functions, the Mellin transforms of newly defined Caputo fractional derivative operator and generating relations for extension of extended hypergeometric functions.

Statistical modeling of lifetime data plays an essential role in a wide range of practical fields, such as health and engineering. There have been a lot of studies done to develop statistical models that can better describe health data than traditional models. For the first time, we pioneer a novel family of continuous probability distributions called the generalized odd beta prime generalized (GOBP-G) family of distributions. The cumulative distribution and probability density functions of the new family are presented. A new generalization of the Weibull distribution called "generalized odd beta prime-Weibull" (GOBPW) is proposed using the pioneered GOBP-G family. The mixture representations of the new distribution are defined and derived. Some formal statistical properties of the GOBPW distribution, such as the moments, moment generating function, incomplete moments, information generating function, entropies, stress-strength function, quantile function, and order statistics, are derived. The estimation of the parameters of the proposed distribution is evaluated using the maximum likelihood estimation approach. Different cancer disease data sets, such as the bladder, head and neck, acute bone, and blood cancers, are used to illustrate the applicability and usefulness of the new model and were compared using several statistical accuracy measures with that of well-established extended Weibull distributions, which are the beta modified Weibull distribution, Kumaraswamy modified Weibull distribution, gamma generalized modified Weibull distribution, gamma log-logistic Weibull distribution, and beta log-logistic Weibull distribution. The results show that the proposed model gives better results than the competitive models. This study could guide the relevant stakeholders in choosing a suitable statistical model for the health data instead of relying on traditional models to enhance decision-making.

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is a novel coronavirus that causes the new Coronavirus disease (COVID-19). The symptoms range from mild to severe with a higher incidence of severe cases seen in patients with risk factors such as older age and comorbidities. COVID-19 is mainly spread through the inhalation of respiratory droplets from coughing or sneezing or via contact with droplet-contaminated surfaces. Paramedics should be aware that some aerosol generating procedures (AGPs) may put them at a higher risk of contracting the virus via possible airborne transmission. The use of remote triage clinical assessment is likely to increase as a result of the pandemic. There is no curative drug treatment for the virus and some medications may exacerbate its effects or make patients more susceptible to it. Paramedics should accept that feeling stressed by the pandemic is a natural response. Official guidelines and advice are evolving continually as the evidence on SARS-CoV-2 and COVID-19 grows. Paramedics should keep up to date with the latest clinical guidance from their employers.

We show how every power series gives rise to a Fibonacci series and a companion series involving Lucas numbers. For illustrative purposes, Fibonacci series arising from trigonometric functions, inverse trigonometric functions, the gamma function and the digamma function are derived. Infinite series involving Fibonacci and Bernoulli numbers and Fibonacci and Euler numbers are also obtained.

Motivated by Kurt's work [Filomat 30 (4) 921-927, 2016], we first consider a class of a new generating function for (p, q)-analog of Apostol type polynomials of order a including Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials of order a. By making use of their generating function, we derive some useful identities. We also introduce (p, q)-analog of Stirling numbers of second kind of order v by which we construct a relation including aforementioned polynomials.

In the paper, the authors establish two identities to express the generating function of the Chebyshev polynomials of the second kind and its higher order derivatives in terms of the generating function and its derivatives each other, deduce an explicit formula and an identities for the Chebyshev polynomials of the second kind, derive the inverse of an integer, unit, and lower triangular matrix, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers respectively with the Chebyshev polynomials, the central Delannoy numbers, and the Fibonacci polynomials.

In the paper, by induction and recursively, the author proves that the generating function of multivariate logarithmic polynomials and its reciprocal are a Bernstein function and a completely monotonic function respectively, establishes a Lévy-Khintchine representation for the generating function of multivariate logarithmic polynomials, deduces an integral representation for multivariate logarithmic polynomials, presents an integral representation for the reciprocal of the generating function of multivariate logarithmic polynomials, computes real and imaginary parts for the generating function of multivariate logarithmic polynomials, derives two integral formulas, and denies the uniform convergence of a known integral representation for Bernstein functions.

In the paper, the author (1) presents an explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials, with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds; (2) recovers an explicit formula and its inversion formula for the Bell polynomials in terms of the Stirling numbers of the first and second kinds, with the aid of the above explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials; (3) constructs some determinantal and product inequalities and deduces the logarithmic convexity of the Bell polynomials, with the assistance of the complete monotonicity of generating functions of the Bell polynomials. These inequalities are main results of the paper.