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This paper presents an efficient solution method for the matrix estimation problem using a static capacity constrained traffic assignment (SCCTA) model with residual queues. The solution method allows for inclusion of route queuing delays and congestion patterns besides the traditional link flows and prior demand matrix whilst the tractability of the SCCTA model avoids the need for tedious tuning of application specific algorithmic parameters. The proposed solution method solves a series of simplified optimization problems, thereby avoiding costly additional assignment model runs. Link state constraints are used to prevent usage of approximations outside their valid range as well as to include observed congestion patterns. The proposed solution method is designed to be fast, scalable, robust, tractable and reliable because conditions under which a solution to the simplified optimization problem exist are known and because the problem is convex and has a smooth objective function. Four test case applications on the small Sioux Falls model are presented, each consisting of 100 runs with varied input for robustness. The applications demonstrate the added value of inclusion of observed congestion patterns and route queuing delays within the solution method. In addition, application on the large scale BBMB model demonstrates that the proposed solution method is indeed scalable to large scale applications and clearly outperforms the method mostly used in current practice.

With some theoretical derivation and practical decision making problem, this study proposes a constructing paradigm for generating fuzzy measures in some certain decision situations, such as in risk evaluation and decision making. The constructing paradigm involves four constructing factors: source measure space, construction function, construction choice, and object/constructed measure space. We use some strict and consistent formulation to reinvent the processes of constructing the well known Belief measure and Plausibility measure. In this study the Belief measure and Plausibility measure are regarded as the two extremities of some pessimism-optimism preferences, and we further propose two construction choices to construct families of fuzzy measures to generalize and seamlessly connect them. We also propose three different ways to determine construction functions in some certain decisional situations. Several relevant mathematical properties and numerical examples are also presented.

In the derivation of the different forms of data assimilation methods there are many mathematical properties are required. This chapter is designed to offer a refresher on mathematical formulas and properties that are important for the understanding of the derivations in this book.

The concept of k-spectrum for genomes is here investigated as a basic tool to analyze genomes. Related spectral notions based on k-mers are introduced with some related mathematical properties which are relevant for informational analysis of genomes. Procedures to generate spectral segmentations of genomes are provided and are tested (under several values of length k for k-mers) on cases of real genomes, such as some human chromosomes and Saccharomyces cerevisiae.

The continuous lognormal distribution has been used to model discrete count data, which is clearly not appropriate. In this paper, we introduce two discrete versions of the continuous lognormal distribution. We study their mathematical properties and estimation issues. Two real data applications show superior performance of the discrete versions over the continuous counter parts.

Linear and circular consecutive models play a vital role to study the mechanical systems emerging in various fields including survival analysis, reliability theory, biological disciplines, and other lifetime sciences. As a result, analysis of reliability properties of consecutive k − out − of − n : F systems has gained a lot of attention in recent years from a theoretical and practical point of view. In the present article, we have studied some important stochastic and aging properties of residual lifetime of consecutive k − out − of − n : F systems under the condition n − k + 1, k ≤ n and all components of the system are working at time t. The mean residual lifetime (MRL) and its hazard rate function are proposed for the linear consecutive k − out − of − n : F (lin/con/k/n:F) and circular consecutive k − out − of − n : F (cir/con/k/n:F) systems. Furthermore, several mathematical properties of the proposed MRL are examined. Finally, the Weibull distribution with two parameters is used as an example to explain the theoretical results.

This paper presents a new Marshall-Olkin extended family of distributions with bounded support. Some of the Mathematical properties of the proposed distribution were studied and the method of maximum likelihood estimation was employed to estimate the unknown parameters of the proposed distribution. A Monte Carlo simulation study was carried out to examine the asymptotic behaviour of the parameter estimates of the distribution.” Finally, two real data sets defined on a unit interval were used to show the applicability of the proposed distribution in analyzing real data sets.

Even though algebraic ideas are addressed across a number of grades, algebra continues to serve as a gatekeeper to upper mathematics and degree attainment because of the high percentage of students that fail algebra classes and become halted in their educational progress. One reason for this is students not having the opportunity to build on their own thinking to connect mathematical ideas from elementary through middle school. The multiplicative field consists of major components of the mathematical ideas learned in these important years, but research within the multiplicative field has focused primarily on the mathematics and not enough on connections within student mathematical thinking. This study focuses on examining the connections in student strategies between whole number, fraction, and two-step rate problems, as well as, how students’ ideas of grouping connect to graphing their strategies. Findings add to previous research of student strategies with multiplication and division by detailing some of the nuance in students’ use of grouping. A focus on grouping strategies reveals students’ progression in understanding the mathematical properties from implicit to explicit to purposeful planning of use of the distributive and associative properties of multiplication. Progression within strategies occurs not as a trajectory, but as part of a constellation of ideas. Additionally, student grouping strategies provided a context to begin to connect solutions to graphing. Implications from this research indicate the need for researchers and teachers to uncover what students know and examine use of grouping to support connections across mathematical concepts within the multiplicative field.

As a typical methodology to deal with uncertain issues, the three-way decision (3WD) has been developed rapidly in nearly ten years, both in theories and applications. Three-way classification is one of the important research fields of 3WD, which utilizes the idea of 3WD to solve classification problems. In this paper, we focus on investigating the effectiveness of three-way classification through two evaluation indicators: the classification quality ( Precision , Recall , Accuracy and F 1 ) and the decision cost. The comparisons between two-way classification and three-way classification are concretely analyzed, some mathematical properties , judging conditions and decision criteria of these two classification methods are also discussed in detail. Finally, the experimental results on eight UCI data sets validate the mathematical analysis, which reveal the effectiveness of three-way classification.

n many applied areas there is a clear need for the extended forms of the well-known distributions.The new distributions are more flexible to model real data that present a high degree of skewness and kurtosis, such that each one solves a particular part of the classical distribution problems. In this paper, a new two-parameter Generalized Odd Gamma distribution, called the (GOGaU) distribution, is introduced and the fitness capability of this model are investigated. Some structural properties of the new distribution are obtained. The different methods including: Maximum likelihood estimators, Bayesian estimators (posterior mean and maximum a posterior), least squares estimators, weighted least squares estimators, Cramér-von-Mises estimators, Anderson-Darling and right tailed Anderson-Darling estimators are discussed to estimate the model parameters. In order to perform the applications, the importance and flexibility of the new model are also illustrated empirically by means of two real data sets. For simulation Stan and JAGS software were utilized in which we have built the GOGaU JAGS module

We propose a new method of adding two parameters to a contin- uous distribution that extends the idea rst introduced by Lehmann (1953) and studied by Nadarajah and Kotz (2006). This method leads to a new class of exponentiated generalized distributions that can be interpreted as a double construction of Lehmann alternatives. Some special models are dis- cussed. We derive some mathematical properties of this class including the ordinary moments, generating function, mean deviations and order statis- tics. Maximum likelihood estimation is investigated and four applications to real data are presented.

Bi-cubic analysis-suitable++ T-splines (AS++ T-splines) (Li in Comput Methods Appl Mech Eng 333:462–474, 2018) are T-splines defined on less restricted T-meshes than analysis-suitable T-splines (AS T-splines), which are both important tools in isogeometric analysis (IGA). In this paper, we generalize the bi-cubic AS++ T-splines to arbitrary degrees and describe some important mathematical properties. Specifically, we develop the conditions under which an AS++ T-spline space belongs to another AS++ T-spline space. This result provides the foundation for the optimized local refinement (Zhang in Comput Methods Appl Mech Eng 342:32–45, 2018) and also is one of the keys for AS++ T-spline approximation. In the end, the optimal approximation properties of the associated T-spline space are developed for arbitrary AS++ T-spline space with the assumption of existence of dual basis, which is automatically satisfied for bi-cubic AS++ T-spline spaces.

Proportional reversed hazard rate distribution family plays an important role in the reliability and lifetime data modelling. In this manuscript this family of distributions is generalized in order to enhance its modelling capability. Considering a special case, we introduce the extended generalized exponential distribution. Its mathematical properties are also studied. New model is applied in fitting levels of TSH hormone and remission times of bladder cancer patients. We believe these results may attract applied statisticians especially those who are in charge with life time data analysis.

The work presents a thermomechanical model for polycrystalline NiTi-based shape memory alloys developed within the framework of generalized standard solids, which is able to cover loading-mode dependent localization of the martensitic transformation. The key point is the introduction of a novel austenite–martensite interaction term responsible for the strain-softening of the material. Mathematical properties of the model are analyzed, and a suitable regularization and a time-discrete approximation for numerical implementation to the finite-element method are proposed. Model performance is illustrated on two numerical simulations: the tension of a superelastic NiTi ribbon and bending of a superelastic NiTi tube.

The process is considered, in which an unreliable rumor spreads in society, which is opposed by the broadcasting of the mass media. In this case, the unreliability of hearing is understood so that the information of the media contains a refutation and thereby inoculates individuals, that is, makes them immune to hearing. At the same time, individuals who have managed to accept the rumor cease to trust the media and thereby become unavailable for persuasion. For this process, a mathematical model is proposed in two versions. The variant with continuous time reveals some of the mathematical properties of the model. The discrete time option is more convenient for analyzing real processes since it allows one to estimate the parameters of the model. To assess these parameters, data on the ratings of the main socio-political programs of Russian TV channels were used. Several scenario calculations of the model with these parameters are presented. The main conclusion is that if the information disseminated by the media is not viral, that is, it is not retold by viewers to their neighbors in society, then the media are unable to resist rumors.

The Sushila distribution is generalized in this article using the quadratic rank transmutation map as developed by Shaw and Buckley (2007). The newly developed distribution is called the Transmuted Sushila distribution (TSD). Various mathematical properties of the distribution are obtained. Real lifetime data is used to compare the performance of the new distribution with other related distributions. The results shown by the new distribution perform creditably well.

Women have consistently lower mortality rates than men at all ages and with respect to most causes. However, gender differences regarding hospital admission rates are more mixed, varying across ages and causes. A number of intuitive metrics have previously been used to explore changes in hospital admissions over time, but have not explicitly quantified the gender gap or estimated the cumulative contribution from cause-specific admission rates. Using register data for the total Danish population between 1995 and 2014, we estimated the time to first hospital admission for Danish men and women aged 60. This is an intuitive population-level metric with the same interpretive and mathematical properties as period life expectancy. Using a decomposition approach, we were able to quantify the cumulative contributions from eight causes of hospital admission to the gender gap in time to first hospital admission. Between 1995 and 2014, time to first admission increased for both, men (7.6 to 9.4 years) and women (8.3 to 10.3 years). However, the magnitude of gender differences in time to first admission remained relatively stable within this time period (0.7 years in 1995, 0.9 years in 2014). After age 60, Danish men had consistently higher rates of admission for cardiovascular conditions and neoplasms, but lower rates of admission for injuries, musculoskeletal disorders, and sex-specific causes. Although admission rates for both genders have generally declined over the last decades, the same major causes of admission accounted for the gender gap. Persistent gender differences in causes of admission are, therefore, important to consider when planning the delivery of health care in times of population ageing. Supplementary Information The online version contains supplementary material available at 10.1007/s10433-021-00614-w.

We introduce a new class of continuous distributions called the Kumaraswamy transmuted-G family which extends the transmuted class defined by Shaw and Buckley (2007). Some special models of the new family are provided. Some of its mathematical properties including explicit expressions for the ordinary and incomplete moments, generating function, Renyi and Shannon entropies, order statistics and probability weighted moments are derived. The maximum likelihood is used for estimating the model parameters. The flexibility of the generated family is illustrated by means of two applications to real data sets.

The degree of statical indeterminacy as a fundamental property in structural mechanics is today mainly known as a property of a complete system without any information about its spatial distribution. The redundancy matrix provides information about the distribution of statical indeterminacy in the system and by this gives an additional valuable insight into the load-bearing behaviour. The derivation and definition of the redundancy matrix are presented based on truss systems and its mathematical properties and their mechanical interpretations are provided. The definition of the redundancy matrix is extended to other discrete systems like beam structures and a definition of the redundancy density is given for the continuous 1D case. Potential applications of the concept include robust design of structures, quantification of imperfection sensitivity as well as assessment of optimal actuator placement in adaptive structures.

A new class of distributions motivated by systems having both series and parallel structures is introduced. Some mathematical properties of the new class (including the moment generating function, moments and order statistics) are derived. Estimation is addressed by the maximum likelihood method and the performance of the estimators assessed by a simulation study. An illustration using three failure data sets shows the usefulness of the new class.

The total and strong version of the Roman domination number (for graphs) is introduced in this research, and the study of its mathematical properties is therefore initiated. We establish upper bounds for such a parameter, and relate it with several parameters concerning vertex domination in graphs. In addition, among other results, we show that for any tree T of order $$n(T)\ge 3$$ , maximum degree $$\Delta (T)$$ and s(T) support vertices, the total strong Roman domination number is bounded below by $$\left\lceil \frac{n(T)+s(T)}{\Delta (T)}\right\rceil +1$$ .

The max–min rodeg ( $${M\!m_{sde}}$$ ) index is a useful topological index in mathematical chemistry. Damir Vuki $$\check{\text {c}}$$ evi $$\acute{\text {c}}$$ studied the mathematical properties of the max–min rodeg index. In this paper, we determine the n-vertex trees with the second, the third and the fourth for $$n\ge 7$$ , and the fifth for $$n\ge 10$$ minimum $$M\!m_{sde}$$ indices, unicyclic graphs with the second and the third for $$n\ge 5$$ , the fourth, the fifth and the sixth for $$n\ge 7$$ , and the seventh for $$n\ge 9$$ minimum $$M\!m_{sde}$$ indices, and bicyclic graphs with the first for $$n\ge 4$$ , the second and the third for $$n\ge 6$$ , and the fourth for $$n\ge 8$$ minimum $$M\!m_{sde}$$ indices.

In this paper, we discuss several mathematical properties and estimation methods for a reparameterized version of the weighted Lindley (RWL) distribution. The RWL distribution can be particularly useful for modeling reliability (survival) data with bathtub-shaped or increasing hazard rate function. The inferential procedure to obtain the parameter estimates is conducted via the maximum likelihood approach considering random right-censoring. Extensive numerical simulations are carried out to investigate and evaluate the performance of the proposed estimation method. Finally, the potentiality of the RWL model is analyzed by employing two real data sets.

Topological index is a numerical value associated with a chemical constitution for correlation of chemical structure with various physical properties, chemical reactivity or biological activity. In this work, some new indices based on neighborhood degree sum of nodes are proposed. To make the computation of the novel indices convenient, an algorithm is designed. Quantitative structure property relationship (QSPR) study is a good statistical method for investigating drug activity or binding mode for different receptors. QSPR analysis of the newly introduced indices is studied here which reveals their predicting power. A comparative study of the novel indices with some well-known and mostly used indices in structure-property modelling and isomer discrimination is performed. Some mathematical properties of these indices are also discussed here.

The generalized gamma-generated family adds one shape parameter to a baseline distribution. We define the gamma-Chen and derive some of its mathematical properties. Its hazard rate may have increasing, decreasing, bathtub and unimodal shapes due to the extra parameter, which portrays a positive point of the proposed model. We perform Monte Carlo simulations to prove that the asymptotic properties of the maximum likelihood estimators hold. We show empirically that the new distribution is better than ten others known distributions using engineering-related data sets.

In this paper we introduce a new technique based on high-dimensional Chebyshev Tensors that we call \emph{Orthogonal Chebyshev Sliding Technique}. We implemented this technique inside the systems of a tier-one bank, and used it to approximate Front Office pricing functions in order to reduce the substantial computational burden associated with the capital calculation as specified by FRTB IMA. In all cases, the computational burden reductions obtained were of more than $90\%$, while keeping high degrees of accuracy, the latter obtained as a result of the mathematical properties enjoyed by Chebyshev Tensors.

This article deals with a model of signal propagation in excitable media based on a system of reaction-diffusion equations. Such media have the ability to exhibit a large response in reaction to a small deviation from the rest state. An example of such media is the nerve tissue or the heart tissue. The first part of the article briefly describes the origin and the propagation of the cardiac action potential in the heart. In the second part, the mathematical properties of the model are discussed. Next, the numerical algorithm based on the finite difference method is used to obtain computational studies in both a homogeneous and heterogeneous medium with an emphasis on interactions of the propagating signals with obstacles in the medium.