Important Mathematical Properties Research Articles

A local-nonlocal mathematical morphology

Mathematical morphology (MM) is traditional and yet applied in many areas. Among the relevant researches, nonlocal extensions have been studied due to their advantages of adaptivity and nonlocal self-similarity. However, these extensions are fragile to noises and easily result in gray value deviation (the maximum grayscale value is changed significantly). In this paper, a local-nonlocal mathematical morphology (LNLMM) is proposed: we use flat structuring element (SE) to avoid gray value deviation and introduce local information to suppress noises. Moreover, to speed up the nonlocal computation involved, we construct the SE in low-dimensional space. Benefiting from the constraint of k-reciprocal nearest neighbors (KRNN) on the SE, the operators of LNLMM theoretically inherit the important mathematical properties from traditional MM, that gives solid supports in applications. With denoising experiments, the powerful performance of LNLMM is preliminarily verified.

Multi-center decomposition of molecular densities: A mathematical perspective.

The aim of this article is to analyze from a mathematical perspective some existing schemes to partition a molecular density into several atomic contributions with a specific focus on Iterative Stockholder Atom (ISA) methods. We provide a unified mathematical framework to describe the latter family of methods and propose a new scheme, named L-ISA (for linear approximation of ISA), which generalizes the so-called additive variational Hirshfeld method. We prove several important mathematical properties of the ISA and L-ISA minimization problems and show that the so-called ISA algorithms can be viewed as alternating minimization schemes, which, in turn, enables us to obtain new convergence results for these numerical methods. Specific mathematical properties of the ISA decomposition for diatomic systems are also presented. Numerical results on diatomic systems illustrate the proven mathematical properties.

Standardized exponentiated Gumbel error innovation distribution in modelling volatility models

Here we proposed three classes of volatility models using Standardized Exponentiated Gumbel Error Innovation Distribution (SEGEID). Important statistical and mathematical properties of this models have been discussed and derived. Hence, the parameters of the volatility models are discussed using the general log likelihood function of the SEGEID. Finally, the volatility models were obtained through partial derivative by adopting a method of numerical method BFGS.

Power Inverted Nadarajah–Haghighi Distribution: Properties, Estimation, and Applications

In this paper, a new three‐parameter power inverted Nadarajah–Haghighi distribution is proposed and studied. The new distribution is a generalization of the inverted Nadarajah–Haghighi distribution in which an additional shape parameter is included. Surprisingly, this additional shape parameter dominates the tails’ weights of distribution and also affects skewness and kurtosis. Such heavy‐tailed distributions are suitable for fitting actuarial and financial risk real problems. Important mathematical properties of this distribution are obtained. Characterization based on two truncated moments is also derived. The parameters are estimated using the maximum likelihood method. Further, a numerical simulation study is conducted for reinforcement of the results. Finally, the application of the proposed distribution to real‐life data sets is presented to prove its applicability.

On the Properties of the New Generalized Pareto Distribution and Its Applications

In this paper, a new generalization of the Generalized Pareto distribution is proposed using the generator suggested in [1], named as Khalil Extended Generalized Pareto (KEGP) distribution. Various shapes of the suggested model and important mathematical properties are investigated that includes moments, quantile function, moment-generating function, measures of entropy, and order statistics. Parametric estimation of the model is discussed using the technique of maximum likelihood. A simulation study is performed for the assessment of the maximum likelihood estimates in terms of their bias and mean squared error using simulated sample estimates. The practical applications are illustrated via two real data sets from survival and reliability theory. The suggested model provided better fits than the other considered models.

A New Extended Cosine—G Distributions for Lifetime Studies

In this article, we introduce a new extended cosine family of distributions. Some important mathematical and statistical properties are studied, including asymptotic results, a quantile function, series representation of the cumulative distribution and probability density functions, moments, moments of residual life, reliability parameter, and order statistics. Three special members of the family are proposed and discussed, namely, the extended cosine Weibull, extended cosine power, and extended cosine generalized half-logistic distributions. Maximum likelihood, least-square, percentile, and Bayes methods are considered for parameter estimation. Simulation studies are used to assess these methods and show their satisfactory performance. The stress–strength reliability underlying the extended cosine Weibull distribution is discussed. In particular, the stress–strength reliability parameter is estimated via a Bayes method using gamma prior under the square error loss, absolute error loss, maximum a posteriori, general entropy loss, and linear exponential loss functions. In the end, three real applications of the findings are provided for illustration; one of them concerns stress–strength data analyzed by the extended cosine Weibull distribution.

The word as a unit of internal predictability

AbstractA long-standing problem in linguistics is how to defineword. Recent research has focused on the incompatibility of diverse definitions, and the challenge of finding a definition that is crosslinguistically applicable. In this study I take a different approach, asking whether one structure is more word-like than another based on the concepts of predictability and information. I hypothesize that word constructions tend to be more “internally predictable” than phrase constructions, where internal predictability is the degree to which the entropy of one constructional element is reduced by mutual information with another element. I illustrate the method with case studies of complex verbs in German and Murrinhpatha, comparing verbs with selectionally restricted elements against those built from free elements. I propose that this method identifies an important mathematical property of many word-like structures, though I do not expect that it will solve all the problems of wordhood.

A new count data model applied in the analysis of vaccine adverse events and insurance claims

Abstract The article presents a new probability distribution, created by compounding the Poisson distribution with the weighted exponential distribution. Important mathematical and statistical properties of the distribution have been derived and discussed. The paper describes the proposed model’s parameter estimation, performed by means of the maximum likelihood method. Finally, real data sets are analyzed to verify the suitability of the proposed distribution in modeling count data sets representing vaccine adverse events and insurance claims.

New Distribution for Fitting Discrete Data: The Poisson-Gold Distribution and Its Statistical Properties

Motivated mainly by lifetime issues, a new lifetime distribution coined ``Discrete Poisson-Gold distribution'' is introduced in this paper. Different structural properties of the new distribution are derived including moment generating function and the $r^{th}$ moment and others are presented. In addition, we discussed various important mathematical properties of the new distribution including estimation procedures for estimating the distribution parameters using the maximum likelihood and method of moments. The usefulness and credibility of the distribution are illustrated by means of two real-data applications to show its superior performance over some other well-known lifetime distributions and to prove its versatility in practical applications.

Scattering matrix pole expansions for complex wave numbers inR-matrix theory

In this follow-up article to [Shadow poles in the alternative parametrization of R-matrix theory, Ducru (2020)], we establish new results on scattering matrix pole expansions for complex wavenumbers in R-matrix theory. In the past, two branches of theoretical formalisms emerged to describe the scattering matrix in nuclear physics: R-matrix theory, and pole expansions. The two have been quite isolated from one another. Recently, our study of Brune's alternative parametrization of R-matrix theory has shown the need to extend the scattering matrix (and the underlying R-matrix operators) to complex wavenumbers. Two competing ways of doing so have emerged from a historical ambiguity in the definitions of the shift $\boldsymbol{S}$ and penetration $\boldsymbol{P}$ functions: the legacy Lane \& Thomas "force closure" approach, versus analytic continuation (which is the standard in mathematical physics). The R-matrix community has not yet come to a consensus as to which to adopt for evaluations in standard nuclear data libraries, such as ENDF. In this article, we argue in favor of analytic continuation of R-matrix operators. We bridge R-matrix theory with the Humblet-Rosenfeld pole expansions, and unveil new properties of the Siegert-Humblet radioactive poles and widths, including their invariance properties to changes in channel radii $a_c$. We then show that analytic continuation of R-matrix operators preserves important physical and mathematical properties of the scattering matrix -- cancelling spurious poles and guaranteeing generalized unitarity -- while still being able to close channels below thresholds.

A Decision-Making Approach Based on New Aggregation Operators under Fermatean Fuzzy Linguistic Information Environment

Fermatean fuzzy linguistic (FFL) set theory provides an efficient tool for modeling a higher level of uncertain and imprecise information, which cannot be represented using intuitionistic fuzzy linguistic (IFL)/Pythagorean fuzzy linguistic (PFL) sets. On the other hand, the linguistic scale function (LSF) is the better way to consider the semantics of the linguistic terms during the evaluation process. It is worth noting that the existing operational laws and aggregation operators (AOs) for Fermatean fuzzy linguistic numbers (FFLNs) are not valid in many situations, which can generate errors in real-life applications. The present study aims to define new robust operational laws and AOs under Fermatean fuzzy linguistic environment. To do so, first, we define some new modified operational laws for FFLNs based on LSF and prove some important mathematical properties of them. Next, the work defines several new AOs, namely, the FFL-weighted averaging (FFLWA) operator, the FFL-weighted geometric (FFLWG) operator, the FFL-ordered weighted averaging (FFLOWA) operator, the FFL-ordered weighted geometric (FFLOWG) operator, the FFL-hybrid averaging (FFLHA) operator and the FFL-hybrid geometric (FFLHG) operator under Fermatean fuzzy linguistic environment. Several properties of these AOs are investigated in detail. Further, based on the proposed AOs, a new decision-making approach with Fermatean fuzzy linguistic information is developed to solve group decision-making problems with multiple attributes. Finally, to illustrate the effectiveness of the present approach, a real-life supplier selection problem is presented where the evaluation information of the alternatives is given in terms of FFLNs. Compared to the existing methods, the salient features of the developed approach are (1) it can solve decision-making problems with qualitative information data using FFLNs; (2) It can consider the attitudinal character of the decision-makers during the solution process; (3) It has a solid ability to distinguish the optimal alternative.

Poincaré sphere of electromagnetic spatial coherence.

We introduce a Poincaré sphere construction for geometrical representation of the state of two-point spatial coherence in random electromagnetic (vectorial) beams. To this end, a novel descriptor of coherence is invoked, which shares some important mathematical properties with the polarization matrix and spans a new type of Stokes parameter space. The coherence Poincaré sphere emerges as a geometric interpretation of this novel formalism, which is developed for uniformly and nonuniformly fully polarized beams. The construction is extended to partially polarized beams as well and is demonstrated with a field having separable spatial coherence and polarization characteristics. At a single point, the coherence Poincaré sphere reduces to the conventional polarization Poincaré sphere for any state of partial polarization.

Remarks on and Characterizations of 2S-Lindley and 2D-Lindley Distributions Introduced by Chesneau et al. (2020)

Chesneau et al.(2020) considered the distributions of sum and differences of two independent and identically distributed random variables with the common Lindley distribution. They derived, very nicely, the above mentioned distributions and provided certain important mathematical and statistical properties as well as simulations and applications of the new distributions. In this short note, we like to show that the assumption of ”independence” can be replaced with a much weaker assumption of ”sub-independence”. Then we present certain characterizations of the proposed distributions to complete, in someway, their work.

A New Distribution for Modeling Extreme Values

In this work, we introduce a new distribution for modeling the extreme values. Some important mathematical properties of the new model are derived. We assess the performance of the maximum likelihood method in terms of biases and mean squared errors by means of a simulation study. The new model is better than some other important competitive models in modeling the repair times data and the breaking stress data.

Impact of pangolin bootleg market on the dynamics of COVID-19 model

In this paper we consider ant-eating pangolin as a possible source of the novel corona virus (COVID-19) and propose a new mathematical model describing the dynamics of COVID-19 pandemic. Our new model is based on the hypotheses that the pangolin and human populations are divided into measurable partitions and also incorporates pangolin bootleg market or reservoir. First we study the important mathematical properties like existence, boundedness and positivity of solution of the proposed model. After finding the threshold quantity for the underlying model, the possible stationary states are explored. We exploit linearization as well as Lyapanuv function theory to exhibit local stability analysis of the model in terms of the threshold quantity. We then discuss the global stability analyses of the newly introduced model and found conditions for its stability in terms of the basic reproduction number. It is also shown that for certain values of R0, our model exhibits a backward bifurcation. Numerical simulations are performed to verify and support our analytical findings.

Potential role of the X circular code in the regulation of gene expression

The X circular code is a set of 20 trinucleotides (codons) that has been identified in the protein-coding genes of most organisms (bacteria, archaea, eukaryotes, plasmids, viruses). It has been shown previously that the X circular code has the important mathematical property of being an error-correcting code. Thus, motifs of the X circular code, i.e. a series of codons belonging to X and called X motifs, allow identification and maintenance of the reading frame in genes. X motifs are significantly enriched in protein-coding genes, but have also been identified in many transfer RNA (tRNA) genes and in important functional regions of the ribosomal RNA (rRNA), notably in the peptidyl transferase center and the decoding center. Here, we investigate the potential role of X motifs as functional elements of protein-coding genes. First, we identify the codons of the X circular code which are frequent or rare in each domain of life (archaea, bacteria, eukaryota) and show that, for the amino acids with the highest codon bias, the preferred codon is often an X codon. We also observe a correlation between the 20 X codons and the optimal codons/dicodons that have been shown to influence translation efficiency. Then, we examined recently published experimental results concerning gene expression levels in diverse organisms. The approach used is the analysis of X motifs according to their density ds(X), i.e. the number of X motifs per kilobase in a gene sequence s. Surprisingly, this simple parameter identifies several unexpected relations between the X circular code and gene expression. For example, the X motifs are significantly enriched in the minimal gene set belonging to the three domains of life, and in codon-optimized genes. Furthermore, the density of X motifs generally correlates with experimental measures of translation efficiency and mRNA stability. Taken together, these results lead us to propose that the X motifs may represent a genetic signal contributing to the maintenance of the correct reading frame and the optimization and regulation of gene expression.

A series solution based formulation for discrete thermal element method of granular materials

This work performs further study on the discrete thermal element method (DTEM) (Feng et al., J. Comput. Phys. 227, 5072–5089 (2008)) of heat conduction in particle systems involving a large number of circular particles in 2D cases. Feng proposed the DTEM based on particular solution for the Laplace equation in a circular domain with Newmann boundary condition. In this paper, another version of DTEM is proposed based on the series solution for the same boundary value problem. The evaluation of the coefficients of the resistance matrix He is a main computational issue associated with the formulation of the DTEM. In this work, another accurate and less time consuming method for calculating He is presented. The resulting equation gives a better expression for hij while in Feng’s work, this was done numerically. An important mathematical property of He is also discovered and discussed. The More Diagonally-Dominant pipe-network model is proposed to make the pipe-network model (Feng et al., Powder Technol. 193, 248–256 (2009)) more accurate. Numerical experiments are conducted to establish the solution accuracy of the proposed method.

Poisson generalized Rayleigh distribution with properties and application

In this study, we have established a new three-parameter Poisson generalized Rayleigh distribution using the Poisson-Generating family of distribution. Some important mathematical and statistical properties of the proposed distribution including probability density function, cumulative distribution function, reliability function, hazard rate function, quantile, median, the measure of skewness, and kurtosis are presented. The parameters of the new distribution are estimated using the maximum likelihood estimation (MLE) method, and constructed the asymptotic confidence intervals also the Fisher information matrix is derived analytically to obtain the variance-covariance matrix for MLEs. All the computations are performed in R software. The capability and applicability of the proposed distribution is exposed by using graphical methods and statistical tests considering a real dataset. We have empirically verified that the Poisson generalized Rayleigh distribution provided a better fit and more flexible in comparison with some selected lifetime distributions.

Forecasts of “normal”

AbstractThe difficulty of forecasting “normal” climate conditions is demonstrated in the context of bivariate normally distributed forecasts and observations. Deterministic and probabilistic skill scores for the normal category are less than for the outer category for all‐but‐perfect models. There are two important mathematical properties of the normal category in a three‐category climatologically equiprobable forecast system that affect the scores for this category. First, the normal category can achieve the highest probability less frequently than the outer categories, and far less frequently in contexts of weak to moderate skill. Second, there are upper limits to the probability the normal category can reach. These mathematical constraints suggest that summary measures of skill may underestimate the predictability and forecast‐skill of extreme events, and that subjective inputs to probabilistic forecasts may need to take greater account of limitations to the predictability of normal conditions.

Versatile mixed methods for the incompressible Navier–Stokes equations

In the spirit of the “Principle of Equipresence” introduced by Truesdell & Toupin, The Classical Field Theories (1960), we use the full version of the viscous stress tensor ν∇u+∇uT−23∇⋅uI which was originally derived for compressible flows, instead of the classical incompressible stress tensor ν∇u. (Note that, here ν is the dynamic viscosity coefficient, and u is the velocity field.) In our approach, the divergence-free constraint for the viscous stress term is not enforced ahead of discretization. Instead, our formulation allows the scheme itself to “choose” a consistent way to interpret the divergence-free constraint: i.e., the divergence-free constraint is interpreted (or enforced) in a consistent fashion in both the mass conservation equation and the stress tensor term (in the momentum equation). Furthermore, our approach preserves the original symmetrical properties of the stress tensor, e.g. its rotational invariance, and it remains physically correct in the context of compressible flows. As a result, our approach facilitates versatility and code reuse. In this paper, we introduce our approach and establish some important mathematical properties for the resulting class of finite element schemes. More precisely, for general mixed methods, which are not necessarily pointwise divergence-free, we establish the existence of a new norm induced by the full, viscous bilinear form. Thereafter, we prove the coercivity of the viscous bilinear form and the semi-coercivity of a convective trilinear form. In addition, we demonstrate L2-stability of the discrete velocity fields for the general class of methods and (by deduction) the H(div)-conforming methods. Finally, we run some numerical experiments to illustrate the behavior of the versatile mixed methods, and we make careful comparisons with a conventional H(div)-conforming scheme.