The asymmetric bimodal exponential power (ABEP) distribution is an extension of the generalized gamma distribution to the real line via adding two parameters which fit the shape of peakedness in bimodality on real line. The special values of peakedness parameters of the distribution are combination of half Laplace and half normal distributions on real line. The distribution has two parameters fitting the height of bimodality, so capacity of bimodality is enhanced by using these parameters. Adding a skewness parameter is considered to model asymmetry in data. The location-scale form of this distribution is proposed. The Fisher information matrix of these parameters in ABEP is obtained explicitly. Properties of ABEP are examined. Real data examples are given to illustrate the modelling capacity of ABEP. The replicated artificial data from maximum likelihood estimates of parameters of ABEP and distributions having an algorithm for artificial data generation procedure are provided to test the similarity with real data.

The main purpose of this paper is to introduce and investigate degenerate Poisson distrib- ution which is a new extension of the Poisson distribution including the degenerate expo- nential function. We then provide several properties of the degenerate Poisson distribution such as the first and the second raw moments and di¤erence operator property. Moreover, we acquired the skewness and the kurtosis for the degenerate Poisson distribution. We also derive its moment generating function by which we define the degenerate Bell polynomials and give a connection for these polynomials related to the unsigned Stirling numbers of the …rst kind.

The goal of clustering is to identify common structures in a data set by forming groups of homogeneous objects. The observed characteristics of many economic time series motivated the development of classes of distributions that can accommodate properties such as heavy tails and skewness. Thanks to its flexibility, the Skewed Exponential Power Distribution (also called Skewed Generalized Error Distribution) ensures a unified and general framework for clustering possibly skewed and heavy tailed time series. This paper develops a clustering procedure of model-based type, assuming that the time series are generated by the same underlying probability distribution but with different parameters. Moreover, we propose to optimally combine the estimated parameters to form the clusters with an entropy weighing $k$-means approach. The usefulness of the proposal is showed by means of application to financial time series, demonstrating also how the obtained clusters can be used to form portfolio of stocks.

It has been empirically observed that the upper tail of the firm size distribution follows either the Pareto distribution or the Zipf distribution, with both patterns being explained by Gibrat’s law (Gabaix, 1999). This article analyzed firm revenue data from China, spanning from 2005 to 2013, to examine the whole range of the firm size distribution. Our empirical analysis revealed that firm revenue data over these years is well fitted by a three-parameter generalized Pareto distribution, with the fitted parameters indicating a dichotomy: The size distribution of large-sized firms, namely the upper tail, is asymptotically characterized by a Pareto distribution or a Zipf distribution, whereas the size distribution of smaller and medium-sized firms is approximated by an exponential distribution. This finding suggests that Gibrat’s law should be extended to account for the emergence of a generalized Pareto distribution.

It has been well known that the exponential distribution is the only continuous distribution that has the memoryless property. Here, we observe that, if the exponential distribution dominates an economy as a probability distribution of income acquisition, then the memoryless property imposes equal opportunities on agents in this economy to acquire earnings. Based on this observation, we propose to identify the emergence of an exponential income distribution as a potential necessary condition for guaranteeing the equality of opportunity. Together with other conditions (such as social and economic mobility), it would promote equal opportunities for income acquisition among citizens. Empirically, we employ the latest data available from four representative market-economy countries (the United Kingdom, the United States, Canada, and China) to demonstrate that the exponential distribution is a stylized feature of the income structure of the low- and middle-income class, which occupies the great majority of populations. By contrast, the top income classes in these countries obey the Pareto distribution. To validate the relationship between exponential distribution and equal opportunity, we empirically show how the income structure of the low- and middle-income class in China (from 1978 to 2015) evolved towards an exponential distribution after the market-oriented economic reformation.

In this paper, we study the well-posedness and asymptotics of a one-dimensional thermoelastic laminated beam system either with or without structural damping, where the heat conduction is given by Fourier's law effective in the rotation angle displacements. We show that the system is well-posed by using Lumer-Philips theorem, and prove that the system is exponentially stable if and only if the wave speeds are equal, by using the perturbed energy method and Gearhart-Herbst-Prüss-Huang theorem. Furthermore, we show that the system with structural damping is polynomially stable provided that the wave speeds are not equal, by using the second-order energy method.

In this paper, we investigate the stabilization of a one-dimensional thermoelastic laminated beam with structural damping, coupled to a heat equation modeling an expectedly dissipative effect through heat conduction governed by Gurtin-Pipkin thermal law. Under some assumptions on the relaxation function g, we establish the well-posedness for the problem. Furthermore, we prove the exponential stability and lack of exponential stability for the problem. To achieve our goals, we make use of the semigroup method, the perturbed energy method and Gearhart-Herbst-Prüss-Huang theorem.

The observed data of COVID-19 progression in Pakistan for first 50 days from the first patient been reported has shown quite an unusual trend which is in opposition to clear exponential spread pattern of any infectious disease. The data of positive cases of 50 days of disease progression has been collected from COVID-19 dashboard of Pakistan and analyzed to see the graphical trend and to forecast the behaviour of disease progression for next 30 days. Mathematical equations regarding exponential growth are used to analyse the disease progression and different possible trajectories are plotted to understand the approximate trend pattern. The possible projections estimated 20k-456k positive cases within 80 days of disease spread in Pakistan. Although, the disease progression pattern is not perfectly exponential, it is still threatening a major fraction of susceptible population and demands effective strategic planning and control.

In this paper, we provide alternative proofs to some results proposed in the article "New inequalities involving circular, inverse circular, hyperbolic, inverse hyperbolic and exponential functions" authored by Yogesh J. Bagul.

As a representative of Exponential Organizations, MI went from obscurity to the Fortune 500 in just ten years. Mi's success is inextricably linked to its outstanding business model. This paper summarizes the elements of MI's business model from four perspectives: value proposition, value creation, value delivery, and value capture, as well as its measures for developing into an exponential organization. Expecting to provide theoretical references for the transformation and upgrading of manufacturing organizations.

Discrete Double Hilbert Exponential sums along polynomials. This is preprint version (not peer reviewed) And we now check the proof of (ii) of Lemma 3.2 whether it is really true.

In this paper we investigate the geometric structure and control of exponential families depending on additional parameters, called external parameters. These generalized exponential families emerges naturally when one applies the maximum entropy formalism to derive the equilibrium statistical mechanics framework. We study the associated statistical model, compute the Fisher metric and introduce a natural fibration of the parameter space over the external parameter space. The Fisher Riemannian metric allows to endow this fibration with an Ehresmann connection and to study the geometry and control of these statistical models. As an example, we show that horizontal lift of paths in the external parameter space corresponds to an isentropic evolution of the system. We apply the theory to the example of a ideal gas in a rotating rigid container. Most of the results are expressed in local coordinates; in the appendices we hint at possible global extensions of the theory.

In previous work (Z. Angew. Math. Phys. 68(2), 2017), Apalara considered a one dimensional thermoelastic laminated beam under Cattaneo’s law of heat conduction and proved the exponential and polynomial decay results depend on the stability number χ_T . In this paper, we continue to study the same system and show that the solution of the concerned system lacks of exponential decay result in the case χ_T ≠ 0 which solves the open problem proposed by Apalara (Z. Angew. Math. Phys. 68(2), 2017).

Although Lov\'{a}sz introduced the graph exponential $G^K$ in 1967, this product has received little attention. This paper focuses on the number of components in $G^K$ when exponent $K$ is a complete graph $K_n$. For a connected bipartite $G$, when $K$ is $K_2$, $G^{K_2}$ has three components; and when $K$ is $K_n$ with $n>2$, $G^{K_n}$ produces $\lceil\frac{n+1}{2}\rceil$ components. A connected $G$ with an odd cycle produces one component in $G^{K_n}$.

This paper shows that the empirical distribution of cross-sectional analyst coverage in China's stock markets follows an exponential law in a given month from 2011 to 2020. The findings hold in both the emerging (Shanghai) and the developed market (Hong Kong). Moreover, the unique distribution parameter (i.e., mean) is directly related to the amount of market-wide information. Average analyst coverage exhibits a significant negative predictive power for stock-market uncertainty, highlighting the role of security analysts in diminishing the total uncertainty. The exponential law can be derived from the maximum entropy principle (MEP). When analysts, who are constrained by average ability in generating information (i.e., the first-order moment), strive to maximize the amount of market-wide information, this objective yields the exponential distribution. Contrary to the conventional wisdom that security analysts specialize in the generation of firm-specific information, empirical findings suggest that analysts primarily produce market-wide information for 25 countries. Nevertheless, it remains unclear why cross-sectional analyst coverage reflects market-wide information, this paper provides an entropy-based explanation.

This paper introduces the exponential time differencing (ETD) technique as a numerical method to solve efficiently American vulnerable options pricing. We address several challenges, including removing cross-derivative terms through appropriate transformations, treating early-exercise opportunities using the penalty method, and substituting fixed boundary conditions with corresponding one-sided finite differences. The proposed method is shown to be both accurate and efficient through numerical experiments, which also compare the results with existing methods and analyse the numerical stability and convergence rate.

This paper presents a new discrete counterpart of the mixture exponential distribution by utilizing the survival discretization method. The moment-generating function and associated moment measures are discussed. The distribution’s hazard rate function can assume increasing or decreasing forms, making it adaptable for diverse fields requiring count data modelling. The paper explores four distinct parameter estimation methods and assesses their performance through Monte Carlo simulations. The applicability of this distribution extends to time series analysis, particularly within the framework of the first-order integer-valued autoregressive process. Additionally, the paper explores quality control applications, addressing serial dependence challenges in count data encountered in production and market management. The performance of two distinct control charts, the cumulative sum chart and the exponentially weighted moving average chart, is evaluated for their effectiveness in detecting shifts in the process means under various models. A bivariate Markov chain approach is used to estimate the average run lengths of these charts, offering valuable insights for implementation. Design recommendations for achieving robustness in-control chart applications are provided. The effectiveness of the proposed models and charts is illustrated using a real data, demonstrating their practical superiority.

Generally, stochastic functional differential equations (SFDEs) pose a challenge as they often lack explicit exact solutions. Consequently, it becomes necessary to seek certain favorable conditions under which numerical solutions can converge towards the exact solutions. This article aims to delve into the convergence analysis of solutions for stochastic functional differential equations by employing the framework of G-Brownian motion. To establish the goal, we find a set of useful monotone type conditions and work within the space Cr((−∞,0];Rn). The investigation conducted in this article confirms the mean square boundedness of solutions. Furthermore, this study enables us to compute both LG2 and exponential estimates.

Proposed in 1889 by Svante Arrhenius, the Arrhenius equation is an important result in physical chemistry which aims to find the relationship between temperature and reaction rates. It is also an essential equation for analytic chemistry which helps to determine the energy barrier for a given reaction. In this article, we will go through a detailed derivation of the exponential factor in the Arrhenius Equation based on the Boltzmann distribution of particle energy probability. Then, we will attempt to model the pre-exponential factor in accordance with the principles of collision theory for bi-molecular reaction with gaseous reactants. Understanding the basic principles of chemical reaction provides valuable tools to analyze complex reaction mechanisms in the perspective of optimizing existing industrial processes.

BackgroundThe novel virus COVID-19, also known as SARS-CoV‑2, is currently rapidly spreading around the globe and pushing healthcare systems to the limits of their capacity. One of the functions of predictive models is to timely act for epidemic preparedness including hospital preparedness. In Egypt, like many other countries in the world, the epidemic situation and forecasting have not yet sufficiently studied. ObjectiveThe study was carried out to develop a short-term forecast scenario for the COVID-19 epidemic situation in Egypt and predict the hospital needs to accommodate the growing number of cases.MethodsSecondary data from the COVID-2019 daily reports and the report issued 8^th of April by the Egyptian Ministry of Health and Population were used. Due to the daily changing level of knowledge and data, the article reflects the status up to 18 April 2020. The prediction was based on the exponential growth rate model. For the depiction of the situation, the full length of the epidemic timeline was analyzed (from February 14^th till April 18^th). The growth rates and their rates of decline during the period from the 22^nd of March till the 18^th of April were calculated and extrapolated in the coming 7 weeks. The predicted hospital needs were assessed against the announced allocated resources.ResultsThe epidemic curve in Egypt is on the ascending arm as of April, 18. The active cases showed exponential growth from the start of the epidemic till April, 18. At the end of this period time, the recovery rate was 23.12% and the case fatality rate (CFR) was7.39. The case fatality rate median level during the last four weeks was 6.64. The active cases are expected to reach more than 20,000 by late May then starts to decline. The allocated regular hospital beds are predicted to show shortage by the time of the release of the paper. The intensive care units (ICU) beds and ventilators are predicted to show insufficiency on May 6.Conclusions: The COVID-19 epidemic in Egypt is expected to continue on the rise for the next few weeks and expected to start to decline late in May, 2020. Our estimates should be useful in preparedness planning. Serious actions should be taken to provide ICU beds and ventilators enough for the predicted number of cases that would need them, not later than the end of April. Mitigation actions have to continue for the coming 6 weeks or until the epidemic situation is more clearly seen.

In this paper, we study the global existence and exponential decay for a dynamic contact problem between a Timoshenko beam with second sound and two rigid obstacles, of which the heat flux is given by Cattaneo's law instead of the usual Fourier's law. The main difficulties arise from the irregular boundary terms, from the low regularity of the weak solution and from the weaker dissipative effects of heat conduction induced by Cattaneo's law. By considering related penalized problems, proving some a priori estimates and passing to the limit, we prove the global existence of the solutions. By considering the approximate framework, constructing some new functionals and applying the perturbed energy method, we obtain the exponential decay result for the approximate solution, and then prove the exponential decay rate to the original problem by utilizing the weak lower semicontinuity arguments.

Introduced in 1967 by Lov\'{a}sz, graph exponentiation has received little attention over the past several decades. Since graph exponentiation produces large graphs and any graph can be constructed utilizing its adjacency matrix, we provide a four step linear construction algorithm that generates the adjacency matrix for the graph exponential.

This research investigates the implementation of complex exponential-based neurons in FPGA, which can pave the way for implementing bio-inspired spiking neural networks to compensate for the existing computational constraints in conventional artificial neural networks. The increasing use of extensive neural networks and the complexity of models in handling big data lead to higher power consumption and delays. Hence, finding solutions to reduce computational complexity is crucial for addressing power consumption challenges. The complex exponential form effectively encodes oscillating features like frequency, amplitude, and phase shift, streamlining the demanding calculations typical of conventional artificial neurons through levering simple phase addition of complex exponential functions. The article implements such a two-neuron and a multi-neuron neural model using Xilinx system generator and Vivado design suite, employing 8-bit, 16-bit, and 32-bit fixed-point data format representations. The study evaluates the accuracy of the proposed neuron model across different FPGA implementations while also providing a detailed analysis of operating frequency, power consumption, and resource usage for the hardware implementations. BRAM-based Vivado designs outperformed Simulink regarding speed, power, and resource efficiency. Specifically, the Vivado BRAM-based approach supported up to 128 neurons, showcasing optimal LUT and FF resource utilization. Such outcomes accommodate choosing the optimal design procedure for implementing spiking neural networks on FPGAs.

Kim-Kim (Russ. J. Math. Phys. 2017, 24, 241-248) defined the degenerate Laplace transform and investigated some of their certain properties. Motivated by this study, in this paper, we introduce the degenerate Sumudu transform and establish some properties and relations. We derive degenerate Sumudu transforms of power functions, degenerate sine, degenerate cosine, degenerate hyperbolic sine, degenerate hyperbolic cosine, degenerate exponential function, and function derivatives. We also acquire a relationship between degenerate Sumudu transform and degenerate gamma function. Moreover, we investigate a scale preserving theorem for the degenerate Sumudu transform. Furthermore, we show that the degenerate Sumudu transform is the theoretical dual transform to the degenerate Laplace transform.

The aim of this study is to investigate, by means of experimental measurements and full-wave simulations, the dominant factors for the very high-resolution (VHR) radar image speckles from exponential correlated rough surfaces. A Ka-band radar system was used to collect the return signal from such a surface sample fabricated by 3D printing, and that signal was further processed into images at different resolution scales, where the image samples were obtained by horizontally turning around the surface sample. To cross-validate the results and to further discuss the VHR speckle properties, full wave simulations by full 3D Finite Difference Time Domain (FDTD) method were conducted with 1600 realizations for the speckle analysis. At the considered very high resolution, speckle statistics show divergence from the fully developed Rayleigh distribution. The factors that impact on the high-resolution speckle properties from exponential correlated rough surface, are analyzed in views of the equivalent number of scatterers theory and scattering scales, respectively. From the data results and extended discussions, it is evident that both of the above factors matter for VHR speckle of backscattering, from the exponential correlated rough surface as a good representative for the ground surface.

In this paper, we study the well-posedness and asymptotic behaviour of solutions to a laminated beam in thermoelasticity of type III. We first give the well-posedness of the system by using the semigroup method. Then, we show that the system is exponentially stable under the assumption of equal wave speeds. Furthermore, it is proved that the system is lack of exponential stability for case of nonequal wave speeds. In this regard, a polynomial stability result is proved.

Information is physically measurable as a selection from a set of possibilities, the domain of information. This defines the term "information". The domain of the information must be known together reproducibly beforehand. As a practical consequence, digital information exchange can be made globally efficient, interoperable and searchable to a large extent by online definition of application-optimized domains of information. There are even more far-reaching consequences for physics. The purpose of this article is to present prerequisites and possibilities for a physical approach that is consistent with the precise definition of information. This concerns not only the discretization of the sets of possible experimental results, but also the order of their definition along time. The access to or comparison with the domain of information is the more frequent, the earlier it was defined. The geometrical appearance of our space is apparently a delayed statistical consequence of a very frequent connection with the common primary domain of information.

It is well established the complex exponential and logarithm are multivalued functions, both failing to maintain most identities originally valid over the positive integers domain. Moreover the general case of complex logarithm, with a complex base, is hardly mentionned in mathematic litterature. We study the exponentiation and logarithm as binary operations where all operands are complex. In a redefined complex number system using an extension of the C field, hereafter named E, we prove both operations always produce single value results and maintain the validity of identities such as log_u (w v) = log_u (w) + log_u (v) where u, v, w in E. There is a cost as some algebraic properties of the addition and subtraction will be diminished, though remaining valid to a certain extent. In order to handle formulas in a C and E dual number system, we introduce the notion of set precision and set truncation. We show complex numbers as defined in C are insufficiently precise to grasp all subtleties of some complex operations, as a result multivaluation, identity failures and, in specific cases, wrong results are obtained when computing exclusively in C. A geometric representation of the new complex number system is proposed, in which the complex plane appears as an orthogonal projection, and where the complex logarithm and exponentiation can be simply represented. Finally we attempt an algebraic formalization of E.

In this note we study the computation of the minimum polynomial of a matrix $A$ and how we can use it for the computation of the matrix $A^n$. We also describe the form of the elements of the matrix $A^{-n}$ and we will see that it is closely related with the computation of the Drazin generalized inverse of $A$. Next we study the computation of the exponential matrix and finally we give a simple proof of the Leverrier - Faddeev algorithm for the computation of the characteristic polynomial.

This paper considers estimating the ratio of two distributions with different parameters and common supports. We consider a Bayesian approach based on the Log--Huber loss function which is resistant to outliers and useful to find robust M-estimators. We propose two different types of Bayesian density ratio estimators and compare their performance in terms of Bayesian risk function with themselves as well as the usual plug-in density ratio estimators. Some applications such as classification and divergence function estimation are addressed.

This paper includes some new investigations and results for post quantum calculus, denoted by (p,q)-calculus. A chain rule for (p,q)-derivative is developed. Also, a new (p,q)-analogue of the exponential function is introduced and some its properties including the addition property for (p,q)-exponential functions are investigated. Several useful results involving (p,q)-binomial coefficients and (p,q)-antiderivative are discovered. At the final part of this paper, (p,q)-analogue of some elementary functions including trigonometric functions and hyperbolic functions are considered and some properties and relations among them are analyzed extensively.

This paper is a part of the broader research on “Adaptation of Laws in Science” and explores the limitations and challenges involved in adapting mathematical concepts to describe natural reality in physics. Specifically, the analysis focuses on two major examples: Euler's exponential laws and spacetime dilation derived from Lorentz transformations. By examining these concepts, the paper emphasizes that some complex laws are not universally applicable for deeper studies of nature, as they originate from mathematics and are correct only for specific parts of nature to which they have been adapted. These complex laws are particularly unsuitable where there is limited information available for adaptation. Therefore, without appropriate adaptations and a deep understanding of the physical context under investigation, the direct use of complex laws can lead to incorrect results. It is essential to avoid excessive generalization of these laws, once they become part of physics, unless a corresponding adaptation to the research context can be made. Furthermore, this paper supports the correctness of relativity theories by invoking that, like Euler's exponential growth theory, they were initially mathematical concepts. However, the nature of mathematics coincided with that of the phenomena in our universe, although through certain adaptations.

In this paper we study a class of impulsive delayed neural networks with parameter uncertainties and reaction-diffusion terms. Using the properties of M-matrices and some inequality techniques, some sufficient conditions are derived to guarantee the global exponential stability of the given system. The results obtained are presented by simple algebraic inequalities, which are certainly more concise than previous methods. Finally, the validity and usefulness of our results are illustrated by arithmetic examples.

The adaptive identification method developed to evaluate the parameters of the Bouc-Wen hysteresis (BWH). The adaptive approach based on the use of adaptive observers. We synthesize adaptive identification algorithms using the second Lyapunov method. The boundedness of adaptive system processes shown in coordinate and parametric spaces. We prove exponential dissipativity of processes in an adaptive system. Estimating method proposed for signaling uncertainty in the system.

This study provides a thorough review and analysis of the evolution of the Greek vehicle fleet over the last ~30 years, which is next used for the generation of high granularity fleet projections and for the estimation of relevant environmental benefits by 2030. The integrated methodology developed takes also into account vehicle clustering and the Brown’s Double Simple Exponential Smoothing technique that together with the adoption of COPERT based emission factors allow for the estimation of the anticipated emissions in 2030. Expected 2030 emissions levels suggest a reduction across all pollutants in comparison to 2018, ranging from 3.7% for PM10 to 54.5% for NMVOC (and 46% for CO, 14% for SO2, 28% for NOX and 21% for CO2). We find that Greece is on track with national goals concerning the reduction of air pollution from the transportation sector, stressing the positive contribution of EVs and new, "greener" vehicles, and setting new challenges for the further improvement of the sector beyond the 2030 outlook.

This paper provides attempts to formalize Hayek’s theory of knowledge. It has been theoretically shown that exponential income distribution is a spontaneous order of the well-functioning market economy. We show that this theoretical result is supported by the empirical evidence from the United Kingdom and China. In particular, we empirically show how the income structure of China evolved towards an exponential distribution after the market-oriented economic reformation. Furthermore, we strictly prove that, if the income structure of an economy obeys an exponential distribution, the income summation over all households leads to an aggregate production function with Hicks-neutral-like technical progress, in which the technology factor is exactly equal to society’s information stock that is a result of combining all of decentralized decisions.

This article investigates the features of heat and mass transfer for the steady two-dimensional Williamson nanofluid flow across an exponentially stretched surface depending on suction/injection. The boundary conditions incorporate the impacts of the Brownian motion and thermophoresis boundary. The analysis of heat transfer is carried out for the two cases of prescribed exponential order surface temperature (PEST) and prescribed exponential order heat flux (PEHF). The ongoing flow problem is mathematically modeled under the basic laws of motion and heat transfer. The similarity variables are allowed to transmute the governing equations of the problem into a similarity ordinary differential equation (ODEs). The solution of this reduced non-linear system of ODEs is supported by the Homotopy analysis method (HAM). The combination of HAM arrangements is acquired by plotting the h-curve. In order to evaluate the influence of several emergent parameters, the outcomes are presented numerically and are plotted diagrammatically as a consequence of velocity, temperature and concentration proles.

In this paper, Lyapunov-Razumikhin technique, design of state-dependent switching laws, a fixed point theorem and variational methods are employed to derive the existence and the unique existence results of (globally) exponentially stable positive stationary solution of delayed reaction-diffusion cell neural networks under Dirichlet zero boundary value, including the global stability criteria \textbf{in the classical meaning}. Next, sufficient conditions are proposed to guarantee the global stability invariance of ordinary differential systems under the influence of diffusions. New theorems show that the diffusion is a double-edged sword in judging the stability of diffusion systems. Besides, an example is constructed to illuminate that any non-zero constant equilibrium point must be not in the phase plane of dynamic system under Dirichlet zero boundary value, or it must lead to a contradiction. Next, under Lipschitz assumptions on active function, another example is designed to prove that the small diffusion effect will cause the essential change of the phase plane structure of the dynamic behavior of the delayed neural networks via a Saddle point theorem. Finally, a numerical example illustrates the feasibility of the proposed methods.

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 Thailand automotive industry grapples with complex challenges in service parts management, demanding a strategic approach for post-sales operations. This research unveils a tailored methodology, leveraging historical sales data, refined part classifications, and strategic decision rules. A high-level overview of the automotive supply chain emphasizes interconnected stakeholders. The systematic methodology delves into part categorization, decision rules, and data analysis. Decision rule outcomes, exceptions, and contingency solutions showcase efficacy. A cost impact assessment highlights substantial savings. Visualization tools offer nuanced perspectives. Optimized service parts classification, robust stocking decision rules, and cost-effective strategies emerge. Support Vector Regression excels in forecasting, with recommendations for dynamic stocking. Implications extend to the broader industry, offering efficiency and service quality blueprints. Acknowledging limitations and suggesting future research, this study contributes a valuable framework for Thai automotive service parts management.

Magnetorheological elastomers (MREs) are a type of smart material that can change their mechanical properties in response to external magnetic fields. These unique properties make them ideal for various applications, including vibration control, noise reduction, and shock absorption. This paper presents an approach for modeling the impact behavior of MREs. The proposed model uses a combination of exponential functions arranged in a multi-layer Preisach model to capture the nonlinear behavior of MREs under impact loads. The model is trained using particle swarm optimization (PSO) and validated using experimental data from drop impact tests conducted on MRE samples under various magnetic field strengths. The results demonstrate that the proposed model can accurately predict the impact behavior of MREs, making it a useful tool for designing MRE-based devices that require precise control of their impact response. The model's response closely matches the experimental data, with a maximum prediction error of 10% or less. Furthermore, the interpolated model's response shows good agreement with the experimental data, with a maximum percentage error of less than 8.5%.

The main result of this work is the estimation of the entropy mode accompanying a wave disturbance, observed at the atmosphere heights range of 90-120km. The study is the direct continuation and development of recent results on diagnosis of the acoustic wave with the separation on direction of propagation. The estimation of the entropy mode contribution relies upon the measurements of the three dynamic variables (the temperature, density and vertical velocity perturbations) of the neutral atmosphere measured by the method of the resonant scattering of radio waves on the artificial periodic irregularities of the ionospheric plasma. The measurement of the atmosphere dynamic parameters has been carried out on the SURA heating facility. The mathematical foundation of the mode separation algorithm is based on the dynamic projecting operator technique. The operators are constructed via the eigenvectors of the coordinate evolution operator of the transformed system of balance equations of the hydro-thermodynamics.

In this paper, we focus on variational Bayesian learning deterministic optimization methods for inference in biparametric exponential models where the parameters follow semiparametric regression structures. This combination of data models and algorithms contributes to solving real-world problems and reduces the computation time. This allows both the rapid exploration of many data models and the accurate estimation of the mean and variance functions through the connection between generalized linear models and graph theory. A simulation study was carried out to assess the performance of the deterministic approximation. Finally, herein, we present an application using macroeconomic data to emphasize the benefits of the proposed approach.

Over the last two decades, residential buildings have accounted for nearly 50 percent of total energy use in New Zealand. In order to reduce household energy use, the factors that influence energy use should be continuously monitored and managed. Building researchers and professionals have made efforts to investigate the factors that affect energy use. However, few have concentrated on the association between household energy use and the cost of residential buildings. This study examined the correlation between household energy use and residential building cost. Analysis of the correlation between energy use data and residential building cost indicated that residential building cost in the construction phase and energy use in the operation stage were significantly correlated. These findings suggest that correct monitoring of building costs can help to identify trends in energy use. Therefore, this study proposes a time series model for forecasting residential building costs of five categories of residential building (one-story house, two-story house, townhouse, apartment, retirement village) in New Zealand. The primary contribution of this paper is the identification of the close correlation between household energy use and residential building costs and provide a new area for optimize energy management.

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.

Generalized Bayes is a Bayesian approach based on the learning rate parameter η. In this study, we examine the effect of parameter η on the estimation results considering joint type-I and type-II hybrid censored samples from k exponential populations. In addition to the learning rate parameter, we consider two loss functions, the Linex and general entropy loss functions in the Bayesian approach. Monte Carlo simulations are performed to compare the performances of the estimation results under losses and different values of η. An illustrative example is performed to study the effect of the learning rate parameter and the different losses with different parameters.

As an on-going pandemic caused by the out-break of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) or simply COVID-19 sweeps through the globe at an unprecedented rate leaving behind trails of high infection and mortality, it is crucial to understand the propagation dynamics of the virus in a host population in order to take urgent and effective remedial and mitigating steps to save life. It is already observed in many countries and communities that accurate and timely testing, tracing, and tracking of the infection lead to better containment and slowing down of the spread. In this exploratory research, the early growth dynamics of infection within a population is pursued based on real data. The study posits that the early growth in a homogenous population follows an exponential pattern motivating further rigorous quantitative treatment based on a number of analytical models such as logistic model, Richard’s model, and Gompertz model– the acceleration pattern of the outbreak is ascertained from the daily inflection data, and regression analysis against population models yields dynamic growth indices which allow very accurate prediction of the successive outbreak size when calibrated continually with updated data. The performance of the various models is evaluated with the real dataset. More, the basic reproduction number of the COVID-19 virus propagation in the community is estimated based on the on-set phase dataset using multi-compartmental epidemiological model. Also, the maximum infection size, infection doubling time and the scope of the herd immunity are also inferred for COVID-19 in a population.

This work offers a new log generalized odd log-logistic exponential regression model for analyzing weekly dengue fever cases in 2022 with a location-systematic component. To achieve this, a data set of 49 observations of dengue fever cases in the Federal District of Brazil is employed. A review of the mathematical properties of the generalized odd log-logistic exponential distribution is provided, the maximum likelihood method is used to estimate the parameters, and, through Monte Carlo simulations, the accuracy of the estimators is investigated. The model's fit is assessed using global influence metrics and residual analysis. For the time scenario studied, the proposed regression identified factors that have an impact on dengue fever cases, which may contribute to improve disease control. Finally, several interpretations are addressed, and a discussion presents results that aid in better understanding the data set and future research on different data.

In this paper, we exhibit new dynamical properties of the two-mode Caudrey-Dodd-Gibbon (TMCDG) equation. This model describes the motion of dual-waves in dispersive and nonlinear media, which mainly depends on the nonlinearity and dispersion parameters. The overlapping of such two-mode waves is affected by the phase-velocity parameter. The study takes a full advantages of the Kudryashov method and of the exponential-expansion method. For the first time, dual-wave solutions are obtained for arbitrary values of the nonlinearity and dispersive factors. The graphs of the novel solutions are also provided in order to depict the waves propagation, as well as to show the influence of the parameters.

This paper aims to quantify the effect of the deadly novel coronavirus (COVID-19) pandemic outbreak on Chinese stock market performance. Shanghai Stock Exchange Composite Index and its component sector indices are examined in this study. The pandemic is represented by a lockdown dummy, new COVID-19 cases and a dummy for 3 February 2020. First, descriptive analysis is performed on these indices to compare their performances before and during the lockdown period. Next, regression analysis with Exponential Generalized Autoregressive Conditional Heteroscedasticity specification is estimated to quantify the pandemic effect on the Chinese stock market. This paper finds that health care, information technology and telecommunication services sectors were relatively more pandemic-resistant, while other sectors were more severely hurt by the pandemic outbreak. The extent to which each sector was affected by pandemic and sentiments in other financial and commodity markets were reported in details in this paper. The findings of this paper are resourceful for investors to avoid huge loss amid pandemic outburst and the China Securities Regulatory Commission in handling future pandemic occurrence to cool down excessive market sentiments.

On 9 March 2020, two cases of COVID-19 were reported in Burkina Faso. As of 10 April 2020, a total number of 484 cases (404 cases in the Kadiogo province) were reported nationwide. Real-time forecasts of COVID-19 are important to inform decision-making in the country. Here, we propose an approach that tests the performance of four models (Exponential Growth model, the Generalized Growth model (GGM), the Generalized Logistic Growth, and Richards Growth model) to select the model that best fit data and to generate short-term forecasting (5-, 10-, and 15-day forecasts from 11 to 25 April 2020) in Kadiogo, the epicenter of the outbreak. Using daily number of confirmed COVID-19 cases, the results suggests that GGM performed the best out of the 4 models. Overall, our GGM predictions suggested an average total number of cumulative cases of 514 (95% CI, 464–559), 629 (95% CI, 559–691), and 750 (95% CI, 661–840) between 11 to 15 April, 16 to 20 April, and 20 to 25 April 2020, respectively. COVID-19 in this province was best approximated by sub exponential growth rather than exponential or logistic growth. Current data suggest that COVID-19 cases would continue to increase over the next 15-days.

Empirical models help us understand the process of plant residue decomposition and nutrient release into the soil. The objective of this study was to determine an appropriate model to describe the decomposition of hairy vetch (Vicia villosa Roth) and cereal rye (Secale cereale L.) cover crop (CC) residue and nitrogen (N) release. Data pertaining to above and belowground CC residue mass loss and N release for up to 2633 cumulative decomposition degree days (112 d) after litterbag installation were obtained from two cropping system experiments, one conducted in 2015 and the other in 2017 and 2018 at the humid subtropical environment of southern IL, USA. Six exponential and two hyperbolic models were fit to percent mass and N remaining data to find the one with minimum Akaike Information Criterion (AIC) and residual sum of squares. Modified three-parameter single exponential and two- or three-parameter hyperbolic models best met the assumed criteria of selection for above and belowground CC residue, respectively. Fitting a double exponential model to a combined data for percent mass and N remaining, which identified two mass and N pools, a fast and a slow pool with different rate constants. A five-parameter double exponential with an asymptote met the preset criteria and passed all tests for normally distributed population, constant variance, and independence of residuals at α = 0.05 when fit to combined data of hairy vetch shoot mass and N remaining. However, a two-parameter hyperbolic and three-parameter asymptotic hyperbolic model provided the best fit to a combined data of cereal rye shoot mass and N remaining, respectively. Both hyperbolic decay models showed a good fit for belowground mass decomposition and N release for both CCs. Cereal rye had poorer fit than hairy vetch for mass and N remaining of both above and belowground mass. The best-selected decay models can be used to estimate the decomposition and N release rates of hairy vetch and cereal rye above and belowground residue in a similar environment.

The turnover of the RNA molecules is determined by the rates of transcription and RNA degradation. Several methods have been developed to study mRNA turnover since the beginnings of molecular biology. Here we summarize the main methods to measure RNA half-life: transcription inhibition, gene control and metabolic labelling. These methods were used to detect the cellular activity of the mRNAs degradation machinery, including the exo-ribonuclease Xrn1 and the exosome. Less progress has been made in the study of the differential stability of mature RNAs because the different methods have often yielded inconsistent results so that an mRNA considered to be stable can be classified as unstable by another method. Recent advances in the systematic comparison of different method variants in yeast have permitted the identification of the least invasive methodologies that reflect half-lives the most faithfully, which is expected to open the way for a consistent quantitative analysis of the determinants of mRNA stability.

The philosophy of testing statistical hypothesis is a natural consequence and functional extension of mathematical analysis of Probability. Along with the concept of recurrence when applied to random sequences and functions, it leads to the analysis of a priori and posterior which implies testing statistical hypothesis. Testing statistical hypothesis also involves algebraic, functional and dimensional considerations, which are found in the works of Laplace. Aspects of mathematical analysis such as universality of solutions, Laws of Large Numbers, Entropy, Information, and various functional dependencies are the main factors explained in the five properties that lead to implication of testing statistical hypothesis. Various interesting examples with modern scientific significance from genetics, astrophysics, and other areas give methodological access to answers of different problems and phenomena which are involved in the logic of testing statistical hypothesis.

This paper represents the first investigation into the consensus problem of linear time-varying multi-agent systems utilizing an event-triggered communication scheme. First, a general event-triggered consensus control scheme is proposed for a general category of linear time-varying multi-agent systems. Under some suitable assumptions, it is demonstrated that all agents’ states will converge exponentially, with Zeno behaviour being ruled out. Second, the consensus problem in a network of linear time-varying multi-agent systems with a spanning tree is investigated using the proposed control strategy. It demonstrates that the consensus issue for the specified system can be reformulated as a stabilization problem for an error system through a time-varying linear transformation. Then, the event-triggered consensus problem is just a special instance of the general event-triggered consensus problem mentioned above. Finally, to illustrate the efficacy of the event-triggered method proposed in this study, simulation results are shown.

Information geometry (IG) is a fascinating combination of differential geometry and statistics where a Riemannian manifold's structure is applied to a statistical model. It may find numerous fascinating uses in the domains of theoretical neurology, machine learning, complexity, and (quantum) information theory, among others. (IG) aims to provide a differential-geometric perspective on statistical geodesic models' structure. In this case, IG, KD, and J-divergence (JD) are used to define the manifold of the Generalised Brownian Motion (GBM). Consequently, the geodesic equations (GEs) are devised, and GB information matrix exponential (IME) is presented. Moreover, for first time ever, the necessary and sufficient mathematical requirement that characterizes the developability of Generalized Brownian Motion(GBM) manifold is devised. Also, a novel sufficient and necessary conditions which characterizes the regions where the surface describing GBM is minimal is determined. Also, it is shown that GBM has a non-zero 0-Gaussian and Ricci curvatures. Consequently, this advances the establishment of a unified GBM- Relativistic Info-Geometric based analysis.

Purpose of this article is to introduce a modification of Phillips operators on the interval $\left[ \frac{1}{2},\infty \right) $ via Dunkl generalization. This type of modification enables a better error estimation on the interval $\left[ \frac{1}{2},\infty \right) $ rather than the classical Dunkl Phillips operators on $\left[ 0,\infty \right) $. We discuss the convergence results and obtain the degrees of approximations. Furthermore, we calculate the rate of convergence by means of modulus of continuity, Lipschitz type maximal functions, Peetre's $K$-functional and second order modulus of continuity.

Purpose of this article is to introduce a modification of Phillips operators on the interval $\left[ \frac{1}{2}% ,\infty \right) $ via Dunkl generalization. This type of modification enables a better error estimation on the interval $\left[ \frac{1}{2},\infty \right) $ rather than the classical Dunkl Phillips operators on $\left[ 0,\infty \right) $. We discuss the convergence results and obtain the degrees of approximations. Furthermore, we calculate the rate of convergence by means of modulus of continuity, Lipschitz type maximal functions, Peetre's $K$-functional and second order modulus of continuity.

The thermoelectric generator (TEG) stands out in many energy harvesters due to its simple structure, small size, etc. However, previous studies rarely probe into the influence of TEG internal resistances on extracting maximum power from TEG, and tracking efficiency is limited. By analyzing the relationship between the tracking efficiency and the TEG internal resistances, a time exponential rate perturbation and observation (P&amp;O) technology is proposed to achieve maximum power point tracking (MPPT) for a wide resistance range of TEG. Using the time exponential rate P&amp;O, the MPPT circuit observes the power change by comparing PMOS on-time and perturbs the power by adjusting NMOS on-time exponentially. The MPPT circuit is implemented in a 110nm CMOS process. The tracking efficiency keeps a high level from 98.9 to 99.5%. The applicable range of the TEG resistance is from 1 to 12Ω, which is enhanced at least 2.2 times.

This paper aims to address the exponential stability and stabilization problems for a class of delayed nonlinear Markov jump systems under randomly occurring Denial-of-Service (DoS) attack and packet loss. Firstly, the stochastic characteristics of DoS attack and packet loss are depicted by the attack success rate and packet loss rate. Secondly, a Period Observation Window (POW) method and a hybrid-input strategy are proposed to compensate for the impact of DoS attack and packet loss on the system. Thirdly, A Dynamic Event-triggered Mechanism (DETM) is introduced to save more network resources and ensure the security and reliability of systems. Then, by constructing a general common Lyapunov functional and combining with the DETM and other inequality analysis techniques, the less conservative stability and stabilization criteria for the underlying systems are derived. In the end, the effectiveness of our result is verified through a numerical example.

The particulars of stimulus-response experiments done on dynamic biosystems clearly limit what one can learn and validate about their structural interconnectivity (topology), even when collected kinetic output data are perfect (noise-free). As always, available access ports and other data limitations rule. For linear systems, exponential modes, visible and hidden, play an important role in understanding data limitations, embodied in what we call dynamical signatures in the data. We show here how to circumscribe and analyze modal response data in compartmentalizing model structures – so that modal analysis can be used constructively in systems biology model building – for nonlinear (NL) as well as linear biosystems. We do this by developing and exploiting the modal basis for dynamical signatures in hypothetical (perfect) input-output data associated with a structural model – one that includes inputs and outputs explicitly – and for NL as well as linear biosystems. The methodology establishes model dimensionality (size, complexity) from particular data sets; helps select among multiple candidate models (model distinguishability); helps in designing new experiments to extract “hidden” structure; and helps to simplify (reduce) models to their essentials. For NL biosystems, results are not as comprehensive, similarly informative about their dominant dynamical properties, and unified with linear models on invariant 2-dimensional manifolds in phase space. Some automation of these highly technical aspects of biomodeling is also introduced.

This paper presents the design of a robust speed controller for brushless DC motors (BLDCM) under field-oriented control (FOC). The proposed robust controller integrates extension theory (ET) and sliding mode theory (SMT) to achieve robustness. First, the speed difference between the speed command and the actual speed of the BLDCM, along with the rate of change of the speed difference, are divided into 20 interval categories. Then, the feedback speed difference and the rate of change of the speed difference are calculated for their extension correlation with each of the 20 interval categories. The interval category with the highest correlation is used to determine the appropriate control gain for the sliding mode speed controller. This gain adjustment tunes the parameters of the sliding surface in the SMT, thereby suppressing the overshoot of the motor speed. Because the sliding surface reaching law of the sliding mode controller (SMC) adopts the exponential approach law, the system's speed response can quickly follow the speed command in any state and exhibit excellent load regulation response. The simplicity of this robust control method, which requires minimal training data, facilitates easy implementation. Finally, the speed control of the BLDCM is simulated using Matlab/Simulink software, and the results are compared with those of the SMC using the constant speed approach law. The simulation results demonstrate that the proposed robust controller exhibits superior speed command tracking and load regulation response compared to the traditional SMC.

Magnetic nanoparticles are met across many biological species ranging from magnetosensitive bacteria, fishes, bees, bats, rats, birds, to humans. They can be both of biogenetic origin and due to environmental contamination, being either in paramagnetic or ferromagnetic state. The energy of such naturally occurring single-domain magnetic nanoparticles can reach up to 10-20 room k_BT in the magnetic field of the Earth, which naturally led to supposition that they can serve as sensory elements in various animals. This work explores within a stochastic modeling framework a fascinating hypothesis of magnetosensitive ion channels with magnetic nanoparticles serving as sensory elements, especially, how realistic it is given a highly dissipative viscoelastic interior of living cells and typical sizes of nanoparticles possibly involved.

Information geometry is a mathematical framework that analyses the structure of statistical models using concepts from differential geometry. It treats families of probability distributions as manifolds, where the parameters of each model determine the coordinate charts. By applying info-geometric tools, we can gain insights into the characteristics of these models. The approach involves characterizing the queueing system's manifold using information geometry and presenting the exponential of the information matrix. This integration of information geometry with queueing theory provides a novel perspective for analyzing the dynamics of queueing systems, incorporating relativistic and Riemannian concepts. Some applications to E-health are highlighted. Finally, closing remarks and the next phase of research.

Aptamers are versatile oligonucleotide ligands used for molecular recognition of diverse targets. However, application of aptamers to the field of amyloid β-protein (Aβ) has been limited so far. Aβ is an intrinsically disordered protein that exists in a dynamic conformational equilibrium, presenting time-dependent ensembles of short-lived, metastable structures and assemblies that have been generally difficult to isolate and characterize. Moreover, despite understanding of potential physiological roles of Aβ, this peptide has been linked to the pathogenesis of Alzheimer disease, and its pathogenic roles remain controversial. Accumulated scientific evidence thus far highlights undesirable or nonspecific interactions between selected aptamers and different Aβ assemblies likely due to metastable nature of Aβ or inherent affinity of RNA oligonucleotides to β-sheet-rich fibrillar structures of amyloidogenic proteins. Accordingly, lessons drawn from Aβ–aptamer studies emphasize that purity and uniformity of the protein target and rigorous characterization of aptamers’ specificity are important for realizing and garnering the full potential of aptamers selected for recongizing Aβ or other intrinsically disordered proteins. This review summarizes studies of aptamers selected for recognizing different Aβ assemblies and highlights controversies, difficulties, and limitations of such studies.

In this work we identify combinations of technological activities that signal the presence local capabilities in a country to successfully export a product. We use country-level patent and trade data to generate a multi-layer network, and we apply maximization of entropy to generate synthetic data to effectively divide signal from noise. We show that in several sectors the signal far exceed the noise. Our exercise provides robust evidence of the presence of synergies between technologies to explain trade performances in specific markets. This can be highly useful for policy makers, to inform industrial and innovation policies.

High-frequency circuit performance is significantly impacted by impedance variations, particularly within the low-resistance regime. Traditional Ohm’s Law based modeling approaches often fail to accurately predict circuit behavior in these conditions, leading to design inaccuracies and potential system failures. The Standard Ohm’s Law based model’s prediction of infinite current as resistance approaches zero is physically unrealistic and hinders its application in practical scenarios. Despite the recognition of these limitations, existing models have not comprehensively addressed the complex impedance behavior observed in high-frequency circuits. To overcome these challenges, this paper introduces a modified version of the Ohm’s Law incorporating an exponential correction term. The Modified Ohm’s Law’s accuracy was evaluated through simulated experiments across a wide frequency range (1kHz to 1GHz) using various electronic components. The findings demonstrate the superior performance of the modified model in predicting currents under low-resistance and high-current conditions compared to the Standard Ohm’s Law model. By providing finite and accurate current values, the proposed model effectively mitigates the unrealistic infinite current predictions of the standard approach. The enhanced predictive capabilities of the Modified Ohm’s Law hold significant implications for high-frequency circuit design and analysis. Its application can lead to improved performance and reliability in power electronics, telecommunications, and other high-frequency systems. By incorporating non-linear impedance behavior, the model offers a more accurate representation of real-world circuit conditions. Future research should focus on refining the exponential term’s parameters to optimize the model's accuracy across a broader range of applications. Additionally, real-time implementation and hardware validation are essential to assess the model's practical utility in complex circuit environments.

In this paper, the problem of predicting future failure times based on a jointly type-II censored sample from k exponential populations is considered. The Bayesian prediction intervals and point predictors were then obtained. Generalized Bayes is a Bayesian study based on a learning-rate parameter. This study investigated the effects of the learning rate parameters on the prediction results. The loss functions of squared error, Linex, and general entropy were used as point predictors. Monte Carlo simulations were performed to show the effectiveness of the learning rate parameter in improving the results of prediction intervals and point predictors.

The main objective of this paper is to continue the development of activities of basic and applied research related to wind energy and to develop methods of optimal control to improve the performance and production of electrical energy from wind. A new control technique of Double fed induction generator for wind turbine is undertaken through a robust approach tagged nonlinear sliding mode control (SMC) with exponential reaching law control (ERL). The SMC with ERL proves to be capable of reducing the system chattering phenomenon as well as accelerating the approaching process. A nonlinear case numerical simulation test is employed to verify the superior performance of the ERL method over traditional power rate reaching strategy. Results obtained in Matlab/Simulink environment show that the SMC with ERL is more robust, prove excellent performance for the control unit by improving power quality and stability of wind turbine.

In this paper described numerical expansion of natural-valued power function x^n, in point x = x₀ where n, x₀ - natural numbers. Apply- ing numerical methods, that is calculus of finite differences, namely, discrete case of Binomial expansion is reached. Received results were compared with solutions according to Newton’s Binomial theorem and MacMillan Double Bi- nomial sum. Additionally, in section 4 exponential function’s e^x representation is shown.

Background Hokkaido is the northernmost, least populous, and coldest of the Japanese islands. It was the first prefecture to be affected by COVID-19, while Kanagawa is home to one of the most populous areas of Japan, namely the Tokyo metro area. The Japanese government responded early during the pandemic by identifying infected patients, contact tracing, and performing PCR analysis on anyone who was suspected of having been exposed to SARS-CoV-2. The government has also been publishing information about each individual who tested positive for the virus. Both Hokkaido and Kanagawa started recording COVID-19 cases in the winter of 2020 and have detailed records of thousands of patients, thus providing an invaluable resource for the transmission and behavior of the virus. Methods The current study analyzed the COVID-19 registry data from the Hokkaido and Kanagawa prefectures. The Hokkaido registry contained 1,269 cases (674 (53%) females and 595 (47%) males) recorded between February 14 and July 22, 2020. The Kanagawa registry had 3,123 cases (1,346 (43%) females and 1,777 (57%) males. The final data contained a total of 4,392 cases (2,020 (46%) females and 2,372 (54%) males). By leveraging the information on viral transmission paths available in the registry data, we performed exponential random graph model (ERGM) network analysis to examine demographic and symptomological homophilies of the SARS-CoV-2 viral transmission networks. Results We observed age, symptomatic, and asymptomatic homophilies in both prefectures. Furthermore, those patients who contracted the virus through secondary or tertiary contacts were more likely to be asymptomatic than those who contracted it from primary infection cases. The transmission networks showed that transmission occurred significantly in healthcare settings, as well as in families, although the size of the networks was small in the latter. Most of the transmissions stopped at the primary and secondary levels and no transmission beyond quaternary was observed. We also observed a higher level of asymptomatic transmission in Kanagawa than in Hokkaido. Conclusions Symptom homophilies are an important component of COVID-19 and suggest that nuanced genetic differences in the virus may affect its epithelial cell type range and can thus result in the diversity of symptoms seen in individuals infected by SARS-CoV-2. Moreover, environmental variables such as temperature and humidity may also be playing an important role in the overall pathogenesis of the virus.

We study Cartan-Schouten metrics, explore invariant dual connections, and propose them as models for Information Geometry. Based on the underlying Riemannian barycenter and the biinvariant mean of Lie groups, we subsequently propose a new parametric mean for data science and machine learning which comes with several advantages compared to traditional tools such as the arithmetic mean, median, mode, expectation, least square method, maximum likelihood, linear regression. We call a metric on a Lie group, a Cartan-Schouten metric, if its Levi-Civita connection is biinvariant, so every 1-parameter subgroup through the unit is a geodesic. Except for not being left or right invariant in general, Cartan-Schouten metrics enjoy the same geometry as biinvariant metrics, since they share the same Levi-Civita connection. To bypass the non-invariance apparent drawback, we show that Cartan-Schouten metrics are completely determined by their value at the unit. We give an explicit formula for recovering them from their value at the unit, thus making them much less computationally demanding, compared to general metrics on manifolds. Furthermore, Lie groups with Cartan-Schouten metrics are complete Riemannian or pseudo-Riemannian manifolds. We give a complete characterization of Lie groups with Riemannian or Lorentzian Cartan-Schouten metrics. Cartan-Schouten metrics are in abundance on 2-nilpotent Lie groups. Namely, on every 2-nilpotent Lie group, there is a 1-1 correspondence between the set of left invariant metrics and that of Cartan-Schouten metrics.

Aptamers have a well-earned place in therapeutic, diagnostic, and sensor applications, and we now show that they provide an excellent foundation for education, as well. Within the context of the Freshman Research Initiative (FRI) at The University of Texas at Austin, students have used aptamer selection and development technologies in a teaching laboratory to build technical and 21st century skills appropriate for research scientists. One of the unique aspects of this course-based undergraduate research experience is that students develop their own projects, and take ownership of their own science in what would otherwise be a traditional teaching lab setting. Of the many successes, this work includes the isolation and characterization of novel calf intestinal alkaline phosphatase (anti-CIAP) RNA aptamers by an undergraduate researcher. Further, preliminary survey data suggest that students who participate in the aptamer research experience express significant gains in their self-efficacy to conduct research, and their perceived ability to communicate scientific results, as well as organize and interpret data. This work will describe the use of aptamers in an educational setting, highlight the positive student outcomes of the aptamer research experience, and more particularly present the research findings relative to the anti-CIAP aptamer.

The Van der Pol oscillator is a chaotic non-linear system. Small perturbations in initial conditions may result in wildly different trajectories. Because of its chaotic nature, controlling, or forcing, the behavior of a Van der Pol oscillator is difficult to achieve through traditional adaptive control methods. Connecting two Van der Pol oscillators together where the output of one oscillator, the driver, drives the behavior of its partner, the responder, is a proven technique for controlling the Van der Pol oscillator. Deterministic AI (DAI) is an adaptive feedback control method that leverages the known physics of the Van der Pol system to learn optimal system parameters for the forcing function. We assessed the performance of DAI employing three different online parameter estimation algorithms. Our evaluation criteria include mean absolute error (MAE) between the target trajectory and the response oscillator trajectory over time. RLS with exponential forgetting (RLS-EF) had the lowest MAE overall, with a 2.46% reduction in error. However, another method was notable. Least Mean Squares with normalized gradient adaptation (LMS-NG) had worse initial error in the first 10% of the simulation, but after that point had consistently better performance. We found that over the last 90% of the simulation, DAI with LMS-NG had a 48.7% reduction in MAE compared to feedforward alone.

For evaluating probabilities of arbitrary random events with respect to a given multivariate probability distribution, specific techniques are of great interest. An important two-dimensional high risk limit law is the Gauss-exponential distribution whose probabilities can be dealt with based upon the Gauss-Laplace law. The latter will be considered here as an element of the newly introduced family of (p,q)-spherical distributions. Based upon a suitably defined non-Euclidean arc-length measure on (p,q)-circles we prove geometric and stochastic representations of these distributions and correspondingly distributed random vectors, respectively. These representations allow dealing with the new probability measures similarly like with elliptically contoured distributions and more general homogeneous star-shaped ones. This is demonstrated at hand of a generalization of the Box-Muller simulation method. En passant, we prove an extension of the sector and circle number functions.

Information geometry (IG) seeks to characterize the structure of statistical geodesic models from a differential geometric point of view. By considering families of probability distributions as manifolds with coordinate charts determined by the parameters of each individual model, the tools of differential geometry, such as divergences and metric tensors, provide effective means of studying their characteristics. The research undertaken in this paper presents a novel approach to the modelling study of information geometrics of a queueing system. In this context, the manifold of a stable M/G/1queue is characterised from the viewpoint of IG. The Fisher Information matrix (FIM) as well as the inverse of (FIM), (IFIM) of stable M/G/1 queue manifold are devised. In addition to that, new results that uncovered the significant impact of stability of M/G/1 queue manifold on the existence of (IFIM) are obtained. The Kullback’s divergence (KD), and J-divergence (JD).New result has been devised on the significant impact of both server utilization and squared coefficient of variation of the underlying M/G/1 queue manifold on both (KD) and (KD) are devised. Also, it is revealed that stable M/G/1 QM is developable (i.e., has a zero Gaussian curvature) and has a non-zero Ricci Curvature Tensor (RCT). Novel stability dynamics of M/G/1 queue manifold is revealed by discovering the mutual dual impact between the behaviour of (RCT) and the stability and the instability phases of the underlying M/G/1 queue manifold. Furthermore, a new discovery that presents the significant impact of stability of M/G/1 queue manifold and the continuity of the unique representation between M/G/1 queue manifold and Ricci Curvature Tensor (RCT). The information matrix exponential (IME) is devised. It is also shown that the obtained (IME) is unstable. Also, it is shown that stability of the devised (IME) enforces the instability of M/G/1 queue manifold. Unifying IG with Queueing Theory enables the study of dynamics of queueing system from a novel Riemannian Geometric (RG) point of view, leading to the analysis of the stable M/G/1 queue, based on the Theory of Relativity (TR).Extending the study over two new additional divergence measures, namely and together with a complete illustrative numerical results for all these measures including KD, JD. This links Queueing theory, IG with deep machine learning and metric learning. Furthermore, this reveals the revolutionary approach of queue learning. Full analytic study of Gaussian curvatures subject to both Angular and Monge techniques together with the overall stability dynamics impact on these curvatures. Full analytic study of Einestein Tensor and Stress Energy Tensor together with the overall stability dynamics impact on these curvatures. The inclusion of the definitions of Gaussian and Ricci, Ricci scalar curvatures and Einstein Tensor together with their physical interpretations; The proposed novel approach for the pioneer visualization of queueing systems via computational information geometry. The determination of new important links between classical queueing theory and other mathematical disciplines, such as IG, matrix theory Riemannian geometry and the THEORY OF RELATIVITY by providing for first time i) The full detailed derivations of the Gaussian curvature ii) The Ricci curvature tensor and iii) The full physical as well as the geometric interpretation of these new results. The provision of a novel link between Ricci Curvature (RCT) and the stability analysis of the stable M/G/1 QM. The full investigation of the newly introduced QT-IG unifiers together with the impact of stability/ instability of the underlying M/G/1 QM on them. The full investigation of the newly introduced (QIGU) unifiers together with the impact of stability/ instability of the underlying M/G/1 QM on them.