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Dasari Naga Vijay Krishna Department of Mathematics, Narayana Educational Instutions, Machilipatnam, Bengalore, India; vijay9290009015@gmail.com Abstract: In this short paper we study some properties of the lines associated with van Aubel’s theorem in the special case when squares are replaced with equilateral triangles constructed on the sides of an arbitrary quadrilateral.

This paper presents an approach to the Kirchhoff-Love plate theory (KLPT) using the finite difference method (FDM). The KLPT covers the case of small deflections, and shear deformations are not considered. The FDM is an approximate method for solving problems described with differential equations. The FDM does not involve solving differential equations; equations are formulated with values at selected points of the structure. Generally in the case of KLPT, the finite difference approximations are derived based on the fourth-order polynomial hypothesis (FOPH) and second-order polynomial hypothesis (SOPH) for the deflection surface. The FOPH is made for the fourth and third derivative of the deflection surface while the SOPH is made for its second and first derivative; this leads to a 13-point stencil for the governing equation. In addition, the boundary conditions and not the governing equations are applied at the plate edges. In this paper, the FOPH was made for all of the derivatives of the deflection surface; this led to a 25-point stencil for the governing equation. Furthermore, additional nodes were introduced at plate edges and at positions of discontinuity (continuous supports/hinges, incorporated beams, stiffeners, brutal change of stiffness, etc.), the number of additional nodes corresponding to the number of boundary conditions at the node of interest. The introduction of additional nodes allowed us to apply the governing equations at the plate edges and to satisfy the boundary and continuity conditions. First-order analysis, second-order analysis, buckling analysis, and vibration analysis of plates were conducted with this model. Moreover, plates of varying thickness and plates with stiffeners were analyzed. Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, with damping taken into account. In first-order, second-order, buckling, and vibration analyses of rectangular plates, the results obtained in this paper were in good agreement with those of well-established methods, and the accuracy was increased through a grid refinement.

This paper presents an approach to the Timoshenko beam theory (TBT) using the finite difference method (FDM). The TBT covers cases associated with small deflections based on shear deformation considerations, whereas the Euler–Bernoulli beam theory neglects shear deformations. The FDM is an approximate method for solving problems described with differential or partial differential equations. It does not involve solving differential equations; equations are formulated with values at selected points of the structure. The model developed in this paper consists of formulating partial differential equations with finite differences and introducing new points (additional or imaginary points) at boundaries and positions of discontinuity (concentrated loads or moments, supports, hinges, springs, brutal change of stiffness). The introduction of additional points allows satisfying boundary and continuity conditions. First-order, second-order, and vibration analyses of structures were conducted with this model. Efforts, displacements, stiffness matrices, buckling loads, and vibration frequencies were determined. In addition, tapered beams were analyzed (e.g., element stiffness matrix, second-order analysis, and vibration analysis). Finally, the direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, considering the damping. The efforts and displacements could be determined at any time.

This paper presents Chebyshev Type inequalities for the Riemann-Liouville variable-order fractional integral operator using two synchronous functions on the set of real numbers. It is the first result of its kind in the current literature using variable-order Riemann-Liouville fractional integral operator. Some special cases for the result obtained in the paper are discussed.

This paper discusses a possible approach to validate the Goldbach conjucture which states that all even numbers can be expressed as a summation of two prime numbers. For this purpose the paper begins with the concept of successive-addition-of-digits-of-an-integer-number (SADN) and its properties in terms of basic algebraic functions like addition, multiplication and subtraction. This concept of SADN forms the basis for classifying all odd numbers into 3 series- the S1, S3 and S5 series- which comprise of odd numbers of SADN(7,4,1), SADN(3,9,6) and SADN(5,2,8) respectively and follow a cyclical order. The S1 and S5 series are of interest in the analysis since they include both prime and composite numbers while the S3 series exclusively consists of composite numbers. Furthermore, the multiplicative property of SADN shows why composites on the S1 series are derived as products of intra-series elements of the S1 and S5 series while composites on the S5 series are derived as products of inter-series elements of the S1 and S5 series. The role of SADN is also important in determining the relevant series for identifying the combination of primes for a given even number since it shows why such combinations for even numbers of SADN(1,4,7) will be found on the S5 series while those for even numbers of SADN(2,5,8) will lie on the S1 series and both the series have a role to play in identifying the prime number combinations for even numbers with SADN(3,6,9). Thereafter, the analysis moves to calculating the total number of acceptable combinations for a given even number that would include combinations in the nature of two composites (c1+c2), one prime and one composite (p+c) and two primes (p1+p2). A cyclical pattern followed by even numbers is also discussed in this context. Identifying the c1+c2 and p+c combinations and thereafter subtracting them from the total number of combinations will yield the number of p1+p2 combinations. For this purpose the paper discusses a general method to calculate the number of composites on the S1 and S5 series for a given number and provides a detailed method for deriving the number of c1+c2 combinations. The paper presents this analysis as a proof to validate the Goldbach conjecture. Since even numbers can be of SADN 1 to 9 and the relation between nTc (i.e. total number of acceptable combinations) and nc(i.e. number of composites) for all even numbers can either be of nTc > nc or nTc ≤ nc, the paper shows that the Goldbach conjecture is true for both these categories of even numbers. In this manner this analysis is totally inclusive of all even numbers in general terms and since the analysis of every even number is common in methodology but unique in compilation, apart from being totally inclusive, it is also mutually exclusive in nature. This proves that the Goldbach conjecture which states that all even numbers can be expressed as atleast one combination of two prime numbers holds true for all even numbers, across all categories possible. Additionally this approach proves that the identification of p1+p2 combinations which would validate the Goldbach conjecture lies in the identification of c1+c2 combinations.

This paper uses the numerical results of surveys sent to Huelva’s (Andalusia, Spain) households to determine the degree of knowledge they have about the urban water cycle, needs, values, and attitudes regarding water in an intermediary city with low water stress. In previous research, we achieved three different households’ clusters. The first one grouped households with high knowledge of the integral water cycle and a positive attitude to smart devices at home. The second cluster described households with low knowledge of the integral water cycle and high sensitivity to price. The third one showed average knowledge and predisposition to have a closer relationship with the water company. This paper continues with this research line, applying Classification and Regression Trees (CART) to determine which hierarchy of variables/factors/ independent components obtained from the surveys are the decisive ones to predict the range of household water consumption in Huelva. Positive attitudes towards improved cleaning habits for personal or household purposes are the highest hierarchy component to predict the water consumption range. Second in the hierarchy, the variable Knowledge Global Score about the integral urban water cycle, associated with water literacy, also contributes to predicting the water consumption range. Together with the three clusters obtained previously, these results will allow us to design water demand management strategies (WDM) fit for purpose that enable Huelva’s households to use water more efficiently.

Let $(A,\Delta)$ be a {\it weak multiplier Hopf algebra} as introduced in [VD-W3] (see also [VD-W2]). It is a pair of a non-degenerate algebra $A$, with or without identity, and a coproduct $\Delta$ on $A$, satisfying certain properties. If the algebra has an identity and the coproduct is unital, then we have a Hopf algebra. If the algebra has no identity, but if the coproduct is non-degenerate (which is the equivalent of being unital if the algebra has an identity), then the pair would be a multiplier Hopf algebra. If the algebra has an identity, but the coproduct is not unital, we have a weak Hopf algebra. In the general case, we neither assume $A$ to have an identity nor do we assume $\Delta$ to be non-degenerate and so we work with a {\it genuine} weak multiplier Hopf algebra. It is called {\it regular} if its antipode is a bijective map from $A$ to itself. \snl In this paper, we {\it continue the study of weak multiplier Hopf algebras}. We recall the notions of the source and target maps $\varepsilon_s$ and $\varepsilon_t$, as well as of the source and target algebras. Then we investigate these objects further. Among other things, we show that the canonical idempotent $E$ (which is eventually $\Delta(1)$) belongs to the multiplier algebra $M(B\ot C)$ where $B=\varepsilon_s(A)$ and $C=\varepsilon_t(A)$ and that it is a {\it separability idempotent} (as studied in [VD4.v2]). If the weak multiplier Hopf algebra is regular, then also $E$ is a {\it regular} separability idempotent. \snl We also consider {\it special cases and examples} in this paper. In particular, we see how for any weak multiplier Hopf algebra $(A,\Delta)$, it is possible to make $C\ot B$ (with $B$ and $C$ as above) into a new weak multiplier Hopf algebra. In a sense, it forgets the 'Hopf algebra part' of the original weak multiplier Hopf algebra and only remembers the source and target algebras. It is in turn generalized to the case of any pair of non-degenerate algebras $B$ and $C$ with a separability idempotent $E\in M(B\ot C)$. We get another example using this theory associated to any discrete quantum group (a multiplier Hopf algebra of discrete type with a normalized cointegral). Finally we also consider the well-known 'quantization' of the groupoid that comes from an action of a group on a set. All these constructions provide interesting new examples of weak multiplier Hopf algebras (that are not weak Hopf algebras).

The paper proposes a new principle of finding and removing elements of mathematical model, redundant in terms of parametric identification of the model. It allows reducing computational and time complexity of the applications built on the model. Especially this is important for AI based systems, systems based on IoT solutions, distributed systems etc. Besides, the complexity reduction allows increasing an accuracy of mathematical models implemented. Despite the model order reduction methods are well known, they are extremely depended however on the problem area. Thus, proposed reduction principles can be used in different areas, what is demonstrated in this paper. The proposed method for the reduction of mathematical models of dynamic systems allows also the assessment of the requirements for the parameters of the simulator elements to ensure the specified accuracy of dynamic similarity. Efficiency of the principle is shown on the ordinary differential equations and on the neural network model. The given examples demonstrate efficient normalizing properties of the reduction principle for the mathematical models in the form of neural networks.

Successive-addition-of-digits-of-a-number(SADN) refers to the process of adding up the digits of an integer number until a single digit is obtained. Concept of SADN has been occasionally identified but seldom employed in extensive mathematical applications. This paper discusses SADN and its properties in terms of addition, subtraction and multiplication. Further, the paper applies the multiplication-property of SADN to understand the distribution of prime numbers. For this purpose the paper introduces three series of numbers -S1, S3 and S5 series- into which all odd numbers can be placed, depending on their SADN and the rationale of such classification. Extending the analysis the paper explains how composite numbers of the S1 and S5 series can be derived. Based on this discussion it concludes that even as the concept of SADN is rather simple in its formulation and appears as an obvious truism but a profound analysis of the properties of SADN in terms of fundamental mathematical functions reveals that SADN holds a noteworthy position in number theory and may have significant implications for unfolding complex mathematical questions like understanding the distribution of prime numbers and Goldbach-problem.

This paper examines whether classical extensional mereology is adequate for formalizing the whole&ndash, parts relation in quantum chemical systems. Although other philosophers have argued that classical extensional and summative mereology does not adequately formalize whole&ndash, parts relation within organic wholes and social wholes, such critiques often assume that summative mereology is appropriate for formalizing the whole&ndash, parts relation in inorganic wholes such as atoms and molecules. However, my discussion of atoms and molecules as they are conceptualized in quantum chemistry will establish that standard mereology cannot adequately fulfill this task, since the properties and behavior of such wholes are context-dependent and cannot simply be reduced to the summative properties of their parts. To the extent that philosophers of chemistry have called for the development of an alternative mereology for quantum chemical systems, this paper ends by proposing behavioral mereology as a promising step in that direction. According to behavioral mereology, considerations of what constitutes a part of a whole is dependent upon the observable behavior displayed by these entities. Thus, relationality and context-dependence are stipulated from the outset and this makes behavioral mereology particularly well-suited as a mereology of quantum chemical wholes. The question of which mereology is appropriate for formalizing the whole&ndash, parts relation in quantum chemical systems is relevant to contemporary philosophy of chemistry, since this issue is related to the more general questions of the reducibility of chemical wholes to their parts and of the reducibility of chemistry to physics, which have been of central importance within the philosophy of chemistry for several decades. More generally, this paper puts contemporary discussions of mereology within the philosophy of chemistry into a broader historical and philosophical context. In doing so, this paper also bridges the gap between formal mereology, conceived as a branch of formal ontology, and &ldquo, applied&rdquo, mereology, conceived as a branch of philosophy of science.

By means of Clifford Algebra, a unified language and tool to describe the rules of nature, this paper systematically discusses the dynamics and properties of spinor fields in curved space-time, such as the decomposition of the spinor connection, the classical approximation of the Dirac equation, the energy-momentum tensor of spinors and so on. To split the spinor connection into the Keller connection Υμ∈Λ1 and the pseudo-vector potential Ωμ∈Λ3 not only makes the calculation simpler, but also highlights their different physical meanings. The representation of the new spinor connection is dependent only on the metric, but not on the Dirac matrix. Only in the new form of connection can we clearly define the classical concepts for the spinor field and then derive its complete classical dynamics, that is, Newton’s second law of particles. To study the interaction between space-time and fermion, we need an explicit form of the energy-momentum tensor of spinor fields, however, the energy-momentum tensor is closely related to the tetrad, and the tetrad cannot be uniquely determined by the metric. This uncertainty increases the difficulty of deriving rigorous expression. In this paper, through a specific representation of tetrad, we derive the concrete energy-momentum tensor and its classical approximation. In the derivation of energy-momentum tensor, we obtain a spinor coefficient table Sabμν, which plays an important role in the interaction between spinor and gravity. From this paper we find that Clifford algebra has irreplaceable advantages in the study of geometry and physics.

The paper proves that an odd composite integer $N$ can be factorized in at most $O( 0.125u(log_2N)^2)$ searching steps if $N$ has a divisor of the form $2^a{u} +1$ or $2^a{u}-1$ with $a > 1$ being a positive integer and $u > 1$ being an odd integer. Theorems and corollaries are proved with detail mathematical reasoning. Algorithms to factorize the kind of odd composite integers are designed and tested by factoring certain Fermat numbers. The results in the paper are helpful to factorize the related kind of odd integers as well as some big Fermat numbers

There are abundant coal, oil and natural gas resources in "Golden Triangle" energy region. Energy resources development, coal chemical industry, coal power and other industries play an important role in promoting regional economic growth, but facing the insufficient water supply. Under the condition of marginal equilibrium of water resources, the paper firstly analyzes the water utilization, and calculates the potential water-saving amount. Then, the paper constructs a cooperative game model and allocates the newly increased GDP by using the weight distribution method. Finally, the paper puts forward some policy suggestions to reduce the efficiency gap of water utilization among regions.

In this paper, initially a mathematical model is formulated for transient frequency of power system considering time delays which occur while transmitting the control signals in open communication infrastructure. Time delay negligence in a power system leads to improper measurement of frequency variation in power system. The study of impact of time delays on the stability of power system is performed by estimating the decay rate of frequency wave form using Kalman Filter (KF). In power system, there is a possibility of multiple time delays. This paper also focusses on developing Interacting Multiple Model (IMM) Algorithm with multiple model space using Kalman Filter (KF) as state estimator tool. The multiple time delays in power system is considered as multiple model space. The result shows that KF provides better estimate of correct model for a particular input-set. The qualitative properties of Riccati difference equation (RDE) in terms of state error covariance of IMMKF are also analyzed and presented.

This paper deals with the study of the existence and non-existence of solutions of a three-parameter family of nonlinear fractional differential equation with mixed-integral boundary value conditions. We consider the &alpha, Riemann-Liouville fractional derivative, with &alpha, &isin, ( 1 , 2 ] . To deduce the existence and non-existence results, we first study the linear equation, by deducing the main properties of the related Green functions. We obtain the optimal set of parameters where the Green function has constant sign. After that, by means of the index theory, the nonlinear boundary value problem is studied. Some examples, at the end of the paper, are showed to illustrate the applicability of the obtained results.

Concept Knowledge (C-K) theory has been used in engineering and science-based research for more than a decade. Design of an Information Technology (IT) artefact is mostly pragmatic in nature. Design Science Research (DSR) methodology applied and studied in many Information Systems (IS) research. Many sub design decisions involved through the design of an IT artefact from a concept (idea) to a working prototype. A DSR artefact is based on a combination of decisions made during several sub-design stages. Artefacts are built based on the selection of elements in each sub-design space. Recording the design decisions on each sub-design space would be beneficial for future researchers. By knowing the design decisions on each sub-design space, researchers would be able to try different combinations of the design. C-K theory provides the ability to capture the design processes’ several sub-design spaces. In this paper, we discuss the DSR research methodology by looking at the stages proposed in the literature, and the application of C-K theory in an IT-based DSR. This paper also proposed a C-K theory-based protocol called Concept Tree for tracking and reporting artefact design steps. The application of C-K theory in DSR is exhibited using the implementation of the Concept Tree for a prototype design IT artefact.

The main objective of this paper is to find a close link between the adaptive level of significance, presented here, and the sample size. We, statisticians, know of the inconsistency, or paradox, in the current classical tests of significance that are based on p-value statistics that is compared to the canonical significance levels (10%, 5% and 1%): "Raise the sample to reject the null hypothesis" is the recommendation of some ill-advised scientists! This paper will show that it is possible to eliminate this problem of significance tests. The Bayesian Lindley's paradox – "increase the sample to accept the hypothesis" – also disappears. Obviously, we present here only the beginning of a possible prominent research. The intention is to extend its use to more complex applications such as survival analysis, reliability tests and other areas. The main tools used here are the Bayes Factor and the extended Neyman-Pearson Lemma.

This paper develops Bayesian inference in reliability of a class of scale mixtures of log-normal failure time (SMLNFT) models with stochastic (or uncertain) constraint in their reliability measures. The class is comprehensive and includes existing failure time (FT) models (such as log-normal, log-Cauchy, and log-logistic FT models) as well as new models that are robust in terms of heavy-tailed FT observations. Since classical frequency approaches to reliability analysis based on the SMLNFT model with stochastic constraint are intractable, the Bayesian method is pursued utilizing a Markov chain Monte Carlo (MCMC) sampling based approach. This paper introduces a two-stage maximum entropy (MaxEnt) prior, which elicits a priori uncertain constraint and develops Bayesian hierarchical SMLNFT model by using the prior. The paper also proposes an MCMC method for Bayesian inference in the SMLNFT model reliability and calls attention to properties of the MaxEnt prior that are useful for method development. Finally, two data sets are used to illustrate how the proposed methodology works.

Background: In the functional data analysis (FDA), the hybrid or mixed data are scalar and functional datasets. The semi-functional partial linear regression model (SFPLR) is one of the first semiparametric models for the scalar response with hybrid covariates. Various extensions of this model are explored and summarized. Methods: Two first research articles, including “semi-functional partial linear regression model”, and “Partial functional linear regression” have more than 300 citations in Google Scholar. Finally, only 106 articles remained according to the inclusion and exclusion criteria such as 1) including the published articles in the ISI journals and excluding 2) non-English and 3) preprints, slides, and conference papers. We use the PRISMA standard for systematic review. Results: The articles are categorized into the following main topics: estimation procedures, confidence regions, time series, and panel data, Bayesian, spatial, robust, testing, quantile regression, varying Coefficient Models, Variable Selection, Single-index model, Measurement error, Multiple Functions, Missing values, Rank Method and Others. There are different applications and datasets such as the Tecator dataset, air quality, electricity consumption, and Neuroimaging, among others. Conclusions: SFPLR is one of the most famous regression modeling methods for hybrid data that has a lot of extensions among other models.

An approach to estimating the objective function value of minimization maximum lateness problem is proposed. It is shown how to use transformed instances to define a new continuous objective function. After that, using this new objective function, the approach itself is formulated. We calculate the objective function value for some polynomially solvable transformed instances and use them as interpolation nodes to estimate the objective function of the initial instance. What is more, two new polynomial cases, that are easy to use in the approach, are proposed. In the end of the paper numeric experiments are described and their results are provided.

In this paper we study solutions and drift homogenization for a class of viscous lake equations by using the method of semigroups of bounded operators. Suppose that the initial value i.e.,for some Hölder continuous function onwith smooth function value satisfying and Then the initial value problem (2) for viscous lake equations has a unique smooth local strong solution. Using this result we study the drift homogenization for three-dimensional stationary Stokes equation in the usual sense

This paper extends the Stochastic Method of Cuts (SMoC) to approximate of the Macroscopic Fundamental Diagram (MFD) of urban networks and uses Maximum Likelihood Estimation (MLE) method to estimate the model parameters based on empirical data from a corridor and 30 cities around the world. For the corridor case, the estimated values are in good agreement with the measured values of the parameters. For the network datasets, the results indicate that the method yields satisfactory parameter estimates and graphical fits for roughly 50\% of the studied networks, where estimations fall within the expected range of the parameter values. The satisfactory estimates are mostly for the datasets which (i) cover a relatively wider range of densities and (ii) the average flow values at different densities are approximately normally distributed similar to the probability density function of the SMoC. The estimated parameter values are compared to the real or expected values and any discrepancies and their potential causes are discussed in depth to identify the challenges in the MFD estimation both analytically and empirically. In particular, we find that the most important issues needing further investigation are: (i) the distribution of loop detectors within the links, (ii) the distribution of loop detectors across the network, and (iii) the treatment of unsignalized intersections and their impact on the block length.

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be a multiplicatively closed subset of [Formula: see text]. The purpose of this paper is to introduce the concept of weakly [Formula: see text]-primary ideals as a new generalization of weakly primary ideals. An ideal [Formula: see text] of [Formula: see text] disjoint with [Formula: see text] is called a weakly [Formula: see text]-primary ideal if there exists [Formula: see text] such that whenever [Formula: see text] for [Formula: see text], then [Formula: see text] or [Formula: see text]. The relationships among [Formula: see text]-prime, [Formula: see text]-primary, weakly [Formula: see text]-primary and [Formula: see text]-[Formula: see text]-ideals are investigated. For an element [Formula: see text] in any general ZPI-ring, the (weakly) [Formula: see text]-primary ideals are characterized where [Formula: see text]. Several properties, characterizations and examples concerning weakly [Formula: see text]-primary ideals are presented. The stability of this new concept with respect to various ring-theoretic constructions such as the trivial ring extension and the amalgamation of rings along an ideal are studied. Furthermore, weakly [Formula: see text]-decomposable ideals and [Formula: see text]-weakly Laskerian rings which are generalizations of [Formula: see text]-decomposable ideals and [Formula: see text]-Laskerian rings are introduced.

For personalized recommender systems, matrix factorization and its variants have become mainstream in collaborative filtering. However, the dot product in matrix factorization does not satisfy the triangle inequality and therefore fails to capture fine-grained information. Metric learning-based models have been shown to be better at capturing fine-grained information than matrix factorization. Nevertheless, most of these models only focus on rating data and social information, which are not sufficient for dealing with the challenges of data sparsity. In this paper, we propose a metric learning-based social recommendation model called SRMC. SRMC exploits users’ co-occurrence patterns to discover their potentially similar or dissimilar users with symmetric relationships and change their relative positions to achieve better recommendations. Experiments on three public datasets show that our model is more effective than the compared models.

This work addresses the problem of sampling from Gaussian probability distributions by means of uniform samples obtained deterministically and directly from space-filling curves (SFCs), a purely topological concept. To that end, the well-known inverse cumulative distribution function method is used, with the help of the probit function,which is the inverse of the cumulative distribution function of the standard normal distribution. Mainly due to the central limit theorem, the Gaussian distribution plays a fundamental role in probability theory and related areas, and that is why it has been chosen to be studied in the present paper. Numerical distributions (histograms) obtained with the proposed method, and in several levels of granularity, are compared to the theoretical normal PDF, along with other already established sampling methods, all using the cited probit function. Final results are validated with the Kullback-Leibler and two other divergence measures, and it will be possible to draw conclusions about the adequacy of the presented paradigm. As is amply known, the generation of uniform random numbers is a deterministic simulation of randomness using numerical operations. That said, sequences resulting from this kind of procedure are not truly random. Even so, and to be coherent with the literature, the expression ”random number” will be used along the text to mean ”pseudo-random number”.

The present paper proposes a fractal analysis of the Covid-19 dynamics in 45 European countries. We introduce a new idea of using the box-counting dimension of the epidemiologic curves as a means of classifying the Covid-19 pandemic in the countries taken into consideration. The classification can be a useful tool in deciding upon the quality and accuracy of the data available. We also investigated the reproduction rate, which proves to have significant fractal features, thus enabling another perspective on this epidemic characteristic. Moreover, we studied the correlation between two meteorological parameters: global radiation and daily mean temperature and two Covid-19 indicators: daily new cases and reproduction rate. The fractal dimension differences between the analysed time series graphs could represent a preliminary analysis criterion, increasing research efficiency. Daily global radiation was found to be stronger linked with Covid-19 new cases than air temperature (with a greater correlation coefficient -0.386, as compared with -0.318), and consequently it is recommended as the first-choice meteorological variable for prediction models.

A century ago, Sir R.A. Fisher introduced for the first time the concept of variance in biological studies. In this paper, we present a few new, modified, or integrated perspectives of variance that we feel would contribute to future thinking and practice of data science. We do so by focusing on brain and behavioral data, through which we hope one could extrapolate the discussions to other fields and data. Specifically: (1) We define different types of variation. (2) We demonstrate that both classic regression models and advanced statistical methods can be viewed as variance-decomposition methods. (3) We make a distinction between innate and acquired variability, linked through Bayesian updating. (4) We review and illustrate how to extract information from high-dimensional data and how to visualize them. Additionally, we introduce the Neural Law of Large Numbers. (5) We discuss the statistical basis for association, explanation, prediction, and causation, and recommend a strategy that may be useful to check if association-based findings can be raised to causal discoveries. Taken together, to understand the variation of data, one needs creative statistical thinking. Meanwhile, by incorporating insights learned from data, one can begin to design better statistical apparatuses.

In this paper, informational (Shannon) measures of symmetry are introduced and analyzed for patterns built of 1D and 2D shapes. The informational measure of symmetry Hsym(G) characterizes the averaged uncertainty in the presence of symmetry elements from group G in a given pattern, whereas the Shannon-like measure of symmetry Ωsym(G) quantifies the averaged uncertainty of the appearance of shapes possessing a total of n elements of symmetry belonging to group G in a given pattern. Hsym(G1)=Ωsym(G1)=0 for the patterns built of irregular, non-symmetric shapes, where G1 is the identity element of the symmetry group. Both informational measures of symmetry are intensive parameters of the pattern and do not depend on the number of shapes, their size, and the entire area of the pattern. They are also insensitive to the long-range order (translational symmetry) inherent for the pattern. Additionally, informational measures of symmetry of fractal patterns are addressed, the mixed patterns including curves and shapes are considered, the time evolution of Shannon measures of symmetry are examined, the close-packed and dispersed 2D patterns are analyzed, and an application of the suggested measures of symmetry for the analysis of the chemical reaction is demonstrated.

Divergences have become a very useful tool for measuring similarity (or dissimilarity) between probability distributions. Depending on the field of application a more appropriate measure may be necessary. In this paper we introduce a family of divergences we call gamma-divergences. They are based on the convexity property of the functions that generate them. We demonstrate that these divergences verify all the usually required properties, and we extend them to weighted probability distribution. We investigate their properties in the context of kernel theory. Finally, we apply our findings to the analysis of simulated and real time series.

This paper describes a polynomial time algorithm for solving graph isomorphism and automorphism. We introduce a new tree data structure called Walk Length Tree. We show that such tree can be both constructed and compared with another in polynomial time. We prove that graph isomorphism and automorphism can be solved in polynomial time using Walk Length Trees.

The power function distribution is a flexible waiting time model that may provide better fit for some failure data. This paper presents the comparison of the maximum likelihood estimates and the Bayes estimates of two-parameter power function distribution. The Bayes estimates are obtained, using conjugate priors, under five loss functions consist of square error, precautionary, weighted, LINEX and DeGroot loss function. The Gibbs sampling algorithm is proposed to generate samples from posterior distributions and in result the Bayes estimates are computed. The comparison of the maximum likelihood estimates and the Bayes estimates are done through the root mean squared errors. One real-life data set is analyzed to illustrate the evaluation of proposed methods of estimation. Finally, results from the simulation are discussed to assess the performance behavior of the maximum likelihood estimates and the Bayes estimates.

In this paper, the notion of θ*-weak contraction is introduced, which is utilized to prove some fixed point results. These results are helpful to give a positive response to certain open question raised by Kannan and Rhoades on the existence of contractive definition which does not force the mapping to be continuous at the fixed point. Some illustrative examples are also given to support our results. As applications of our result, we investigate the existence and uniqueness of a solution of non-linear matrix equations and integral equations of Volterra type as well.

In this paper, we present the definition, some properties, and solve a problem on special primes. These properties help in providing us with a better understanding of the problem posed related to special primes on the open problem garden website. The problem involves finding all the primes q, given a prime p such that q≡1(mod p) and 2^((q−1)/p)≡1(mod q). We prove that a prime number q is a special prime of p if and only if the order of 2 in U(q) divides q−1p. Also, we prove that a prime number q is not a special prime for any prime number if 2 is a generator of the group U(q) and that there exist infinitely many special primes for any given prime number.

The objective of this paper is to investigate the convergence of coupling-parameter expansion-based solutions to the Ornstein–Zernike equation in liquid state theory. The analytically solved Baxter’s adhesive hard sphere model is analyzed first by using coupling-parameter expansion. It was found that the expansion provides accurate approximations to solutions—including the liquid-vapor phase diagram—in most parts of the phase plane. However, it fails to converge in the region where the model has only complex solutions. Similar analysis and results are obtained using analytical solutions within the mean spherical approximation for the hardcore Yukawa potential. However, numerical results indicate that the expansion converges in all regions in this model. Next, the convergence of the expansion is analyzed for the Lennard-Jones potential by using an accurate density-dependent bridge function in the closure relation. Numerical results are presented which show convergence of correlation functions, compressibility versus density profiles, etc., in the single as well as two-phase regions. Computed liquid-vapor phase diagrams, using two independent schemes employing the converged profiles, compare excellently with simulation data. The results obtained for the generalized Lennard-Jones potential, with varying repulsive exponent, also compare well with the simulation data. Solution-spaces and the bifurcation of the solutions of the Ornstein–Zernike equation that are relevant to coupling-parameter expansion are also briefly discussed. All of these results taken together establish the coupling-parameter expansion as a practical tool for studying single component fluid phases modeled via general pair-potentials.

The present paper deals with a class of even entire functions of order $\rho =1$ and genus $\vartheta =0$ of the polynomials form, \[ \sum\limits_{m=0}^\infty {\frac{\left( {-1} \right)^m\Phi ^{\left( {2m} \right)}\left( 0 \right)}{\Gamma \left( {2m+1} \right)}x^{2m}} =\Phi \left( 0 \right)\prod\limits_{k=1}^\infty {\left( {1-\frac{x}{\ell _k }} \right)} , \] where $\Phi\left( 0 \right)\ne 0$, real numbers $x$, nonnegative integers $m$, and $\ell _k \ne 0$ are all of the nonzero roots with $\sum\limits_{k=1}^\infty {1/\left| {\ell _k } \right|}

In this paper we address some variants for the products of Hadamard and Patterson. We prove that all zeros of the Riemann $\Xi$--function are real. We also prove that the Riemann hypothesis is true. The equivalence theorems associated with the Riemann zeta--function are obtained in detail.

In a recent paper, we have a shown that the flattening of galactic rotation curves can be explained by retardation. However, this will rely on a temporal change of galactic mass. In our previous work, we kept only second order terms of the retardation time in our analysis, while higher terms in the Taylor expansion where not considered. Here we consider analysis to all orders and show that a second order analysis will indeed suffice, and higher order terms can be neglected., 15 pages, 10 figures

In this paper, we introduce a new iteration scheme, named as the S**-iteration scheme, for approximation of fixed point of the nonexpansive mappings. This scheme is faster than Picard, Mann, Ishikawa, Noor, Agarwal, Abbas, Thakur, and Ullah iteration schemes. We show the stability of our instigated scheme and give a numerical example to vindicate our claim. We also put forward some weak and strong convergence theorems for Suzuki's generalized nonexpansive mappings in the setting of uniformly convex Banach spaces. Our results comprehend, improve, and consolidate many results in the existing literature.

In this paper, we prove some common fixed point theorems on complex partial metric space. The presented results gener- alize and expand some of the literature well-known results. We also explore some of the application of our key results.

Since December 2019, the coronavirus disease (COVID-19) has rapidly spread worldwide. The Mexican government has implemented public safety measures to minimize the spread of the virus. In this paper, the authors use statistical models in two stages to estimate the total number of coronavirus (COVID-19) cases per day at the state and national level in Mexico. Two types of models are proposed: first, a polynomial model of the growth for the first part of the outbreak until the inflection point of the pandemic curve and then a second nonlinear growth model is used to estimate the middle and the end of the outbreak. Model selection will be performed using Vuong’s test. The proposed models show overall fit similar to predictive models (e.g. time series, and machine learning); however, the interpretation of parameters is less complex for decision-makers and the residuals follow the expected distribution when fitting the models without autocorrelation being an issue.

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 have motivated the development of classes of distributions that can accommodate properties such as heavy tails and skewness. Thanks to its flexibility, the Skew Exponential Power Distribution (also called Skew Generalized Error Distribution) ensures a unified and general framework for clustering possibly skewed time series. This paper develop 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 all the parameter estimates to form the clusters with an entropy weighing k-means approach. The usefulness of the proposal is showed by means of an application to financial time series, showing also how the obtained clusters can be used to form portfolio of stocks.

In this paper we investigate the global asymptotic stability of following system ofhigher order difference equations with quadratic terms:xn+1=A+Byn/yn−m^2, yn+1=A+Bxn/xn−m^2, where A and B are positive numbers and the initial values are positive numbers.We also study the boundedness, rate of convergence and oscillation behaviour of thesolutions of related system.

This paper proposed a definition of the fractional line integral, generalising the concept of the fractional definite integral. The proposal replicated the properties of the classic definite integral, namely the fundamental theorem of integral calculus. It was based on the concept of the fractional anti-derivative used to generalise the Barrow formula. To define the fractional line integral, the Grünwald–Letnikov and Liouville directional derivatives were introduced and their properties described. The integral was defined for a piecewise linear path first and, from it, for any regular curve.

With the vaccination against Covid-19 now available, how vaccination campaigns influence the mathematical modeling of epidemics is quantitatively explored. In this paper, the standard susceptible-infectious-recovered/removed (SIR) epidemic model is extended to a fourth compartment, V, of vaccinated persons. This extension involves the time t-dependent effective vaccination rate, v(t), that regulates the relationship between susceptible and vaccinated persons. The rate v(t) competes with the usual infection, a(t), and recovery, μ(t), rates in determining the time evolution of epidemics. The occurrence of a pandemic outburst with rising rates of new infections requires k+b, Physics, 3 (2), ISSN:2624-8174

We first of all define the arrow of time. Definition of the arrow of time will allow choosing different initial starting points. One of the issues we will also discuss is the interconnection be-tween the arrow of time, entropy and quantum information. Seth Lloyd in his 2001 work made a linkage between entropy, bits, and information via an axiomatic approach involving time in-tervals. Our take is a bit more general. We will discuss as well the t’Hooft’s statement as to in-itial conditions and times arrow, and how different cosmological models may influence initial conditions. Spoilers alert, if a nonsingular start to expansion existed, this would provide the most straightforward way to avoid a datum from classical relativity. That is, that in the actual equations of classical GR, there is no reason to have time asymmetry. Time asymmetry is built into initial conditions and we will detail several candidates. The first half of the paper brings up cosmology models and forming the arrow of time. The second is related to entropy itself and the problem of information. .

This paper explores multiple perspectives and proofs regarding the validity of what is considered to be one of the – if not THE - most controversial proofs in the field of mathematics, historical and contemporary applications alike. We consider the various explanations and equivalents of the axiom along with the more widely receptive alternatives. Furthermore, we review the resistance the axiom has encountered in various fields and its potential use in the research and expansion of said fields.

The maximization of entropy S within a closed system is accepted as an inevitability (as the second law of thermodynamics) by statistical inference alone. The Maximum Entropy Production Principle (MEPP) states that not only S maximizes, but $\dot{S}$ as well: a system will dissipate as fast as possible. There is still no consensus on the general validity of this MEPP, even though it shows remarkable explanatory power (both qualitatively and quantitatively), and has been empirically demonstrated for many domains. In this theoretical paper I provide a generalization of entropy gradients, to show that the MEPP actually follows from the same statistical inference, as that of the 2nd law of thermodynamics. For this generalization I only use the concepts of super-statespaces and microstate-density. These concepts also allow for the abstraction of 'Self Organizing Criticality' to a bifurcating local difference in this density, and allow for a generalization of the fundamentally unresolved concepts of 'chaos' and 'order'.

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.

In cotton cultivars, an insect that causes irreversible damage is the Spodoptera frugiperda, known as the fall armyworm. Since the visual detection of plants is a burdensome task for human inspection, the spectral information related to plant damage, registered on a spectral scale, can be useful. These measurements, associated with machine learning techniques, produce useful information for a rapid and non-invasive inspection method development. To contribute to this gap fulfillment, this paper proposes a machine learning framework to model the spectral response of cotton plants under the attack of the fall armyworm. Additionally, a theoretical model is presented, built from the results of the machine learning analysis, to infer this damage with up-to-date orbital sensors. The data was composed of the reflectance measurements collected at a cotton field with control plants and plants submitted to Spodoptera frugiperda damage. Their spectral response was recorded with a hand-held spectroradiometer ranging from 350 to 2,500 nm, for eight consecutive days. Different machine learning models were evaluated and the overall best model was defined by accuracies comparisons on a testing-set. A ranking approach was adopted based on the model accuracy, returning the most contributive wavelengths for the classification. Sequentially, an unsupervised neural network (Self-Organizing Map - SOM) was implemented to reduce data-dimensionality and assist in the definition of important spectral regions. The regions were associated with the spectral bands of the two sensors (OLI and MSI) and a theoretical model using a band simulation process with the overall best machine learning model was proposed. The results indicated that the Random Forest (RF) algorithm is the most suitable to predict cotton-plants damaged by insects and that the last day of analysis (8th day) was better to separate it, with F-measure equals 0.912. The ranking approach combined with the SOM method indicated the spectral regions at the red to near-infrared (650 to 1,350 nm) and shortwave infrared (1,570 to 1,640 nm) as the most important regions to the analysis. The proposed theoretical model simulated with the OLI and MSI sensor-bands returned an F-Measure of 0.865 and 0.886, respectively. In conclusion, this framework can be used to map cotton-plants under insect-attack. The theoretical model presents high accuracy to infer the insect-damaged on cotton plants based on multispectral bands from other sensors, being a useful tool for future research that intends to evaluate it in other areas and at different field scales.

In this paper, we investigate the recurrence properties of the generalized Tribonacci sequence and present how the generalized Tribonacci sequence at negative indices can be expressed by the sequence itself at positive indices.