Multiplicative Variables Research Articles

A generalization of the CIRCE method for quantifying input model uncertainty in presence of several groups of experiments

The semi-empirical nature of best-estimate models closing the balance equations of thermal-hydraulic (TH) system codes is well-known as a significant source of uncertainty for accuracy of output predictions. This uncertainty, called model uncertainty, is usually represented by multiplicative (log-)Gaussian variables whose estimation requires solving an inverse problem based on a set of adequately chosen real experiments. One method from the TH field, called CIRCE,22This is a French acronym: Calcul des Incertitudes Relatives aux Corrélations Élémentaires. addresses it. We present in the paper a generalization of this method to several groups of experiments each having their own properties, including different ranges for input conditions and different geometries. An individual (log-)Gaussian distribution is therefore estimated for each group in order to investigate whether the model uncertainty is homogeneous between the groups, or should depend on the group. To this end, a multi-group CIRCE is proposed where a variance parameter is estimated for each group jointly to a mean parameter common to all the groups to preserve the uniqueness of the best-estimate model. The ECME algorithm for Maximum Likelihood Estimation developed in Celeux et al. (2010) is extended to the latter context, then applied to relevant demonstration cases. Finally, it is tested on a practical case to assess the uncertainty of critical mass flow assuming two groups due to the difference of geometry between the experimental setups.

A Stochastic Dynamical Model of Slope Creep and Failure

AbstractWe propose a stochastic dynamical model to simulate slope secondary and tertiary creep phenomena. The slope secondary creep is represented by the Kesten process defined as a stochastic affine auto‐regressive process involving both multiplicative and additive random variables. The Kesten process can realistically capture the co‐existence of a background deformation and intermittent displacement bursts, which are together well characterized by an inverse gamma velocity distribution. The slope tertiary creep is modeled by a nonlinear stochastic dynamical equation embodying a nonlinear feedback mechanism and a nonlinear random effect, which can mimic the development of slow or catastrophic landslides. For catastrophic landslides, the probability density function of slope velocities tends to deviate from the inverse gamma distribution by populating the “dragon‐king” regime, although sometimes they may grow undetectably in the “black‐swan” regime. Our model provides a quantitative framework to understand, simulate, and interpret complex landslide displacement time series.

Multiplicity of Scator Roots and the Square Roots in $\mathbb{S}^{1+2}$

This paper presents the roots of elliptic scator numbers in $\mathbb{S}^{1+n}$, which includes both the fundamental $2\pi$ symmetry and the $\pi$-pair symmetry for $n\geq2$. Here, the scator set $\mathbb{S}^{1+n}$ is a subset of $\mathbb{R}^{1+n}$ with the scator product and the multiplicative representation. These roots are expressed in terms of both additive (rectangular) and multiplicative (polar) variables. Additionally, the paper provides a comprehensive description of square roots in $\mathbb{S}^{1+2}$, which includes a geometrical representation in three-dimensional space that provides a clear visualization of the concept and makes it easier to understand and interpret. Finally, the paper handles whether the aspects should be further investigated.

Finite-Time Fault Detection Under Deception Attacks: A Hybrid Dynamic Variables-Dependent Event-Based Scheme

This article deals with the dynamic event-based finite-time fault detection (FD) issue for singularly perturbed systems under deception attacks. A hybrid dynamic variables-dependent event-triggered mechanism (ETM) is proposed, which contains both additive and multiplicative internal dynamic variables (IDVs). Such a new dynamic ETM (DETM) is employed to decide whether the measurement data packet should be released or not at each time instant. A Bernoulli random variable is utilized to describe the deception attack that occurs during the measurement data packet transmission. Attention is to design an FD filter (FDF) which guarantees that the resultant error dynamics of filtering on FD is stochastically <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> finite-time bounded. Based on a new Lyapunov function containing two IDVs, sufficient conditions are derived to ensure the existence of the FDF whose parameters are designed by resorting to the feasible solutions of several matrix inequalities. Two examples are presented to demonstrate the effectiveness of the DETM-based FDF design method. It is revealed that the devised DETM has superiority over some existing ETMs in saving network resources while not degrading the FD filtering performance.

Frequency domain analysis of pre-stressed elastomeric vibration isolators

Two types of elastomeric vibration isolators used for equipment vibration isolation in aerospace vehicles are considered for the present study. These isolators are constructed using elastomers mounted in steel encasings. These isolators are initially deformed statically and dynamic loads are applied on the deformed configuration. To capture the static deformation, equivalent static load corresponding to its load rating and specified displacements are created. Static deformation is computed using Finite Element methods with four node axi-symmetric element which include the geometric non-linear effect for steel and with standard Yeoh hyper-elastic material model for elastomers (Muhammed and Zu, 2012) [1]. Yeoh material constants are derived from uni-axial tension test data of the elastomer specimen. These isolators are subjected to harmonic and random excitations in the pre-deformed state. For numerical analysis, elastomeric constants at dynamic conditions are obtained as complex function of frequency using Dynamic Mechanical Analyzer (DMA) for a range of frequencies. The standard material model of Yeoh is modified incorporating frequency dependant material characteristics and damping in the range of frequencies of interest. A multiplicative non-separable variables law is derived for Yeoh material model to include the effect of static pre-stress, based on the methodology given in literature (Nashif et al., 1985; Beda et al., 2014) [2,3]. The modifications of Yeoh model suitable for frequency domain analysis is the novelty in the present study. In the analysis, while dynamic loads are applied, the configuration is updated considering initial static loading. The frequency response of the isolators is computed using material properties evaluated at progressive dynamic strains until a match in natural frequency is observed. Appropriate damping corrections are then incorporated to match the test observed transmissibility. Then updated material properties are used to compute the random response which showed good agreement with results of experiments, validating the approach taken for the development of this model.

Consumers' Perception of Food and Agriculture Education in Farmers' Markets in Taiwan.

Following the trend of green product consumption, farmers’ markets sell green products and have gradually developed the promotion of environmentally friendly knowledge of food and agricultural education. However, past research on farmers’ markets has rarely been combined with the function of food and agricultural education. Therefore, this study explores the relationship between product knowledge, green perceived value, and purchase intention from the perspective of farmers’ market food and agricultural education. Furthermore, research and investigation were conducted with trust and local attachment as multiplicative adjustment variables. This study adopted structural equation modeling (SEM) to examine the structural relationship among local attachment, product knowledge, green perceived value, purchase intention, and model fit. We distributed 396 surveys at three large farmers’ markets in Taiwan, of which 88.6% were used in this study. The research results showed that product knowledge and green perceived value had a significant positive impact on purchase intention; trust positively affected product knowledge and purchase intention; the multiplication of local attachment and trust also positively impacted product knowledge and purchase intention. Based on the research results of this study, it is suggested that while transferring knowledge, the farmers’ markets can incorporate trust and emotional relationships, which will serve as stimulating factors that can increase consumers’ purchase intentions. Although the findings are local to Taiwan, its characteristics are typical of food and agricultural education, as well as farmers’ markets worldwide, especially in Asia.

Group decision making based on consistency and consensus analysis of dual multiplicative linguistic preference relations

This paper proposes a new group decision making (GDM) method based on the consistency and the consensus analysis of dual multiplicative linguistic preference relations (DMLPRs). A new type of linguistic variables , called dual multiplicative linguistic variables (DMLVs), is presented, which is defined on the multiplicative linguistic scale. DMLVs are used to represent asymmetrical qualitative hesitancy judgments of decision makers (DMs). A maximum-consistency-based interactive algorithm to derive multiplicative linguistic intuitionistic preference relations (MLIPRs) is presented, by which the consistency concept for DMLPRs is obtained. Then, we define the concept of inconsistent DMLPRs and propose an optimal-model-based method for deriving consistent DMLPRs. Furthermore, incomplete DMLPRs also can be dealt with by the proposed maximum-consistency-based interactive algorithm. For GDM, the weights of DMs are determined by the cosine-based correlation coefficient between individual DMLPRs. Moreover, we propose a consensus measure to calculate the agreement degree of DMLPRs and build an optimal model to increase the consensus level of individual DMLPRs. Finally, a new GDM method (call Algorithm III ) is offered and an application example is used to illustrate the proposed GDM method. The proposed GDM method outperforms the former GDM methods for GDM in the environments of DMLPRs.

Interpreting spatial regression models with multiplicative interaction explanatory variables

Interpreting spatial regression models with multiplicative interaction explanatory variables

Combinatorial decompositions for monomial ideals

It is well known that Riquier introduced the notion of multiplicative variables applied to order ideals to represent initial conditions of partial differential equations as series. On the other hands, Janet introduced the concept of involutive division. Following Riquier and Janet, we focus on the following problem.Suppose one needs to compute a monomial ideal generated in some degree D and its escalier, knowing the Hilbert function and some monomials which must belong to the ideal/escalier. We give combinatorial tools to answer this question. First we define combinatorial decompositions of sets of terms, showing the criteria to decompose the set T≥D of the terms in degree greater or equal than D, into disjoint subsets called cones. This is done by assigning to each term t of degree D some multiplicative variables. The cone of t is then the set formed by t and all its multiples obtained multiplying it by all possible products of powers of multiplicative variables. Then, supposed one has selected a decomposition, we deal with the escalier/ideal partition problem, namely we study the rules which force a term to be in the ideal (or in the escalier) provided that another term does, so that both the ideal and the escalier turn out to be decomposed in disjoint cones by the decomposition set on the whole T≥D.

Maurice Janet’s algorithms on systems of linear partial differential equations

This article describes the emergence of formal methods in theory of partial differential equations (PDE) in the French school of mathematics through Janet’s work in the period 1913–1930. In his thesis and in a series of articles published during this period, Janet introduced an original formal approach to deal with the solvability of the problem of initial conditions for finite linear PDE systems. His constructions implicitly used an interpretation of a monomial PDE system as a generating family of a multiplicative set of monomials. He introduced an algorithmic method on multiplicative sets to compute compatibility conditions, and to study the problem of the existence and the uniqueness of a solution to a linear PDE system with given initial conditions. The compatibility conditions are formulated using a refinement of the division operation on monomials defined with respect to a partition of the set of variables into multiplicative and non-multiplicative variables. Janet was a pioneer in the development of these algorithmic methods, and the completion procedure that he introduced on polynomials was the first one in a long and rich series of works on completion methods which appeared independently throughout the twentieth-century in various algebraic contexts.

Learning Tracking Control Over Unknown Fading Channels Without System Information.

A novel data-driven learning control scheme is proposed for unknown systems with unknown fading sensor channels. The fading randomness is modeled by multiplicative and additive random variables subject to certain unknown distributions. In this scheme, we propose an error transmission mode and an iterative gradient estimation method. Unlike the conventional transmission mode where the output is directly transmitted back to the controller, in the error transmission mode, we send the desired reference to the plant such that tracking errors can be calculated locally and then transmitted back through the fading channel. Using the faded tracking error data only, the gradient for updating input is iteratively estimated by a random difference technique along the iteration axis. This gradient acts as the updating term of the control signal; therefore, information on the system and the fading channel is no longer required. The proposed scheme is proved effective in tracking the desired reference under random fading communication environments. Theoretical results are verified by simulations.

A rational and consensual method for group decision making with interval-valued intuitionistic multiplicative preference relations

Interval-valued intuitionistic multiplicative variables (IVIMVs) can conveniently and effectively represent the uncertain multiplicative preferred and non-preferred judgements of decision makers, this paper studies group decision making (GDM) with interval-valued intuitionistic multiplicative preference relations (IVIMPRs). To calculate the interval-valued intuitionistic multiplicative priority weight vector reasonably, a new consistency concept is introduced that satisfies robustness and upper triangular property. Following this concept, models to judge the consistency and obtain consistent IVIMPRs from inconsistent ones are constructed, respectively. To address the problem of incomplete preferences, consistency-based model to determine missing values is built. Furthermore, we study the consensus for GDM with IVIMPRs and provide a new consensus approach. When an acceptable consensus level is not achieved, an interactive and automatic adjustment method is applied to reach a better consensus level. Following discussion about consistency and consensus, an algorithm for GDM is offered that can address inconsistent and incomplete IVIMPRs. Finally, a practical problem about selecting the steel supplier is selected to show the application of the new method.

Restoring incomplete PUMLPRs for evaluating the management way of online public opinion

In the age of big data explosion, the management of online public opinion has encountered great challenges. Which way can effectively manage online public opinion has become a decision-making question for us to think about. Probabilistic uncertain multiplicative linguistic preference relations (PUMLPRs) are a remarkable instrument to solve uncertain evaluation problems. This paper uses the PUMLPRs to assess the management ways of the online public opinion. Owing to the intricacy of decision-making domain, the PUMLPRs are not always complete. We get the complete PUMLPRs in two steps: the repair for uncertain multiplicative linguistic variables and the repair for probability. Moreover, the consistency of the complete PUMLPRs is researched. Then the final priorities are obtained by the proposed possibility degree formula. After that, the numerical example that helps assess the valid way to manage online public opinion is performed to check the feasibility of the proposed decision-making procedure.

Group Decision Making with Interval-Valued Intuitionistic Multiplicative Linguistic Preference Relations

To express the asymmetrically uncertain preferred and non-preferred qualitative judgments of decision makers, this paper introduces interval-valued intuitionistic multiplicative linguistic variables (IVIMLVs). To show their application in decision making, a ranking method is first offered. Then, we introduce IVIMLVs for preference relations and propose interval-valued intuitionistic multiplicative linguistic preference relations (IVIMLPRs). To obtain the ranking reasonably, a consistency definition for IVIMLPRs is presented. A mathematical optimization model for judging the consistency of IVIMLPRs based on the new concept is constructed. To address two general cases: incompleteness and inconsistency, mathematical optimization models for ascertaining unknown values in incomplete IVIMLPRs and deriving completely consistent IVIMLPRs from inconsistent ones are built, respectively. For group decision making, a consensus index is defined to measure the consensus achieved among the decision makers’ preferences. If the consensus is not enough, a mathematical optimization model for improving the consensus level is established. Furthermore, a linear optimization model for determining the weights of the decision makers based on the consensus analysis is constructed. Finally, a group decision-making method with IVIMLPRs based on consistency and consensus analysis is offered, and its application on selecting supply chain cooperative partners is offered.

Group decision making with multiplicative interval linguistic hesitant fuzzy preference relations

This paper proposes multiplicative interval linguistic hesitant fuzzy sets (MILHFSs) to denote the asymmetrically qualitative judgments with uncertain and hesitant characteristics, which are expressed by several interval linguistic judgments defined on the asymmetrical linguistic term set. Considering the utilization of MILHFSs, the concept of multiplicative interval linguistic hesitant fuzzy variables (MILHFVs) is introduced, and multiplicative interval linguistic hesitant fuzzy preference relations (MILHFPRs) are proposed. Then, we analyze the consistency and introduce an approach for computing the priority vector from consistent MILHFPRs. Meanwhile, incomplete MILHFPRs are discussed, and a model for obtaining unknown MILHFVs following the analysis of consistency is built. Considering group decision making (GDM), the analysis of consensus is made. Subsequently, an approach for GDM with inconsistent and incomplete MILHFPRs is provided that follows the analysis of consistency and consensus. The feasibility of the developed results is demonstrated by two practical GDM problems: (1) assessing seven-seat sport utility vehicles (SUVs) and (2) selecting the public-private partnership (PPP) model.

A Novel Group Decision Making Method Under Uncertain Multiplicative Linguistic Environment for Information System Selection

An effective method is proposed to solve the group decision-making problems with uncertain multiplicative linguistic information. First, to compare two uncertain multiplicative linguistic variables, a possibility degree formula is introduced in accordance with the operation laws. Some desirable properties of possibility degree are provided. Then, the uncertain multiplicative linguistic hybrid weighted geometric averaging (ULHWG) operator is developed to aggregate uncertain multiplicative linguistic variables into an overall preference value. Thus, a comprehensive algorithm for linguistic group decision-making is proposed based on the deviation measure and the ULHWG operator. Finally, a numerical example is provided to illustrate the practicality and validity of the proposed method.

Calculating priority weights from interval-valued multiplicative hesitant fuzzy preference relations

To denote the reciprocally uncertain hesitancy of decision makers, this paper introduces a new type of fuzzy sets called interval-valued multiplicative hesitant fuzzy variables (IVMHFVs), which is an extension of multiplicative hesitant fuzzy variables. On the basis of IVMHFVs, interval-valued multiplicative hesitant fuzzy preference relations (IVMHFPRs) are introduced. To rank objects based on IVMHFPRs reasonably, the consistency of IVMHFPRs is studied and a consistency definition is presented. Using this concept, a 0–1 mixed programming model for judging the consistency of IVMHFPRs is built, and a consistency-based programming model for deriving consistent IVMHFPRs is constructed. When the offered IVMHFPRs are incomplete, a model for determining missing values is established. In the setting of group decision making, a consensus index is defined and an interactive method for improving the consensus is offered. After that, an approach to group decision making with inconsistent and incomplete IVMHFPRs is developed. Meanwhile, a practical example is provided to show the application of the new method.

Acceleration of Poor Women Capability in Increasing Income Through Self-Help Group

This study analyzed the impact of the acceleration of women capabilities in creating family income through self-help group based micro credit. The analysis unit of this study are the members of UP2K-PKK (Efforts to Increase Family Income). They are poor women who run productive business. The samples of this research are 177 members of the UP2K-PKK in West Semarang. The data is collected using a questionnaire that is delivered directly to the respondents along with in-depth interview. The analysis tool used is multiple linear regressions. This research tests the model that is the family income as the dependent variable and the development capability through self-help group based micro credit as the independent variable. The acceleration of capability through self-help group based micro credit includes microcredit, training, partnerships, technical assistance , membership. The poor women have individual capabilities (initial endowment) including education, business experience, and membership. To know that the capability development through self-help group accelerates the development of individual capabilities, then the multiplicative variables are inserted.The results of this research show that capability development through self-help group accelerates the impact of micro credit and initial endowment in creating the family income.

Moments of Random Multiplicative Functions and Truncated Characteristic Polynomials

We give an asymptotic formula for the $2k$th moment of a sum of multiplicative Steinhaus variables. This was recently computed independently by Harper, Nikeghbali and Radziwill. We also compute the $2k$th moment of a truncated characteristic polynomial of a unitary matrix. This provides an asymptotic equivalence with the moments of Steinhaus variables. Similar results for multiplicative Rademacher variables are given.

An approach to multiplicative linguistic group decision making based on possibility degrees

AbstractIn order to compare two uncertain multiplicative linguistic variables, a possibility degree formula has been developed and its properties are also studied. Then a possibility degree matrix (also considered as a reciprocal preference relation) is constructed to compare a collection of uncertain multiplicative linguistic variables. Two models are further provided to derive the priority vector from the possibility degree matrix based on the additive consistency and multiplicative consistency. Especially, if the parameters are assigned specific values, then the models reduce to the existing ones. A group decision making method has been developed to deal with the situations where the preferences on alternatives are expressed by uncertain multiplicative linguistic variables. In this method, the possibility degree matrix is first constructed, from which the priority of alternatives is obtained using the developed models. At the end, an example is given to illustrate the proposed method.