نشریه پژوهشهای ریاضی
سال هشتم شماره 4 (پیاپی 23، زمستان 1401)

In this paper we consider two type of mathematical models for the novel coronavirus (2019-nCov), which are in the form of a nonlinear differential equations system. In the first model the contact rate, , and transition rate of  symptomatic infected indeviduals to the quarantined infected class, , are constant. And in the second model these quantities are time dependent. These models are the SEIR one, where are Susceptible, Exposed, Infected and Recovered classes of human population respectively. We establish the Newton-Taylor polynomial solutions for these system, so that the nonlinear systems are solvable by an iterative and progressive process with a good accuracy. We completely describe the algorithm of such systems in another paper and here we express briefly. This algorithm action on the interval , where  is the length of partial intervals, and  is the number of intervals. In every partial interval, we linearize the problem by the Newton's method and then solve the linear problem by the Taylor polynomial solutions technique. We extensively investigate the numerical analysis of the method.

Improving the quality, compliance rate and product life is a priority in manufacturing industries. If the performance of a process is evaluated by the lifetime of a product, it is clear that a larger lifetime indicates better product quality and higher process capability, and this process is accepted. For instance, the lifetime of electronic components exhibits a larger-the-better type of quality characteristic. The process capability index is one of the indicators for evaluating the capability of a process.  To measure the larger-the-better quality characteristic, a process capability index has been defined as CL=μ-Lσ ,  0≤L<∞,  where  μ  and σ  are the mean and standard deviation of the process, respectively,  and L  is the lower specification limit.In various manufacturing and service processes, the assumption of normality is often not valid. The two-parameter exponential distribution has many applications in engineering, biological studies, epidemiology, and medical sciences. In engineering analysis, the location parameter is called the threshold value or warranty time and the scale parameter is the expected average as well as warranty life. If the lifetime of a manufactured product follows a two-parameter exponential distribution with location parameter θ  and scale parameter λ , then the capability index becomes CL=1-1λL-θ .

Since the performance capability index has been used in industries, point estimation, construction of confidence intervals, and testing the hypothesis about this index have been considered. However, the classical inference cannot be applied to CL .  Therefore, we utilized the concepts of generalized pivotal quantity and generalized test variable for inference on this parameter. Here, we suppose that different kinds of schemes such as double type II censoring, progressive censoring, record values, and k-record values. In each case, the maximum likelihood and uniformly minimum variance unbiased estimators are derived. Then, a generalized confidence interval and a generalized p-value are suggested.  The generalized confidence interval is evaluated by Monte Carlo simulation, and the presented approaches are illustrated using some real examples.

We study the coverage probability and expected length of the proposed generalized confidence interval for CL  under a progressive censoring scheme. We found that the coverage probability is close to the confidence coefficient. Also, it does not depend on the values of parameters and sample size. Besides, the expected length decreases when λ  decreases (or the sample size increases).  The shortest expected length of the generalized confidence interval is when the value of L  is close to the location parameter θ .

The following conclusions were drawn from this research.The minimum variance unbiased estimator for CL  has a closed-form under all considered schemes.The proposed generalized inference for CL  is a satisfactory and easy approach.This article contains all results in other papers such as Lee et al. (2011) and Gunasekera, and Wijekularathna (2019).Different indicators have been introduced to evaluate the process’s capability. Some of the most important ones that are widely used in manufacturing industries have been introduced by Wooluru et al. (2014). The methods introduced in this paper, can be generalized and used for these process capability indicators.

In this Article, proposes an approximation for the solution of the Riccati equation based on the use of exponential spline functions. Then the exponential spline equations are obtained and the differential equation of the fractional Riccati is discretized. The effect of performing this mathematical operation is obtained from an algebraic system of equations. To illustrate the benefits of the method proposed here the error analysis and convergence article are also discussed based on this exponential spline. Finally, a second-order method is obtained. One of the benefits of this proposed method is that it is not only for solving fractional Riccati equations, but also for a variety of Fractional equations that can be used. To illustrate the effectiveness of this method by solving numerical examples and a comparison of the results obtained from the implementation of this proposed method with the results of other existing numerical methods proves the claim that the proposed method is a good approximation for the fractional Riccati equations.

In this paper, based on variational methods and critical point theory, we guarantee the existence of infinitely many classical solutions for a two-point boundary value problem with fourth-order Sturm-Liouville equation; Some recent results are improved and by presenting one example, we ensure the applicability of our results.

In this paper, we are intend to present a numerical algorithm for computing approximate solution of linear and nonlinear Fredholm, Volterra and Fredholm-Volterra  integro-differential equations. The approximated solution is written in terms of fractional Jacobi polynomials. In this way, firstly we define Riemann-Liouville fractional operational matrix of fractional order Jacobi polynomials, then by using this matrix and the least squares method the solution of equation reduce to a system of algebraic equations which is solved through the Newton’s iterative method. In the next step we analyze convergence of the solution, and then to confirm the theoretical issue we examine some numerical examples. The results indicate the accuracy and efficiency of the method. The excellence of this method is its generality, which includes the fractional order Legendre and Chebyshev polynomials. Also it is also easy to use for linear and nonlinear integro-differential equations and provides good results.

Invariance principles is one of the ways to summarize sample information and by these principles invariance or equivariance decision rules are used. In this paper, first, the methods for finding the maximal invariant function are introduced. As a new method, maximal invariant statistics are constructed using equivariant functions. Then, using several equivariant functions, the maximal invariant statistic is obtained. If the group acts uniquely transitively, the equivariant functions will be converted into estimators that can be used in statistical branches. Finally, it is shown that in the case that the group action is not uniquely transitive but a normal subgroup has this property, the maximal invariant statistic can also be calculated.

In this article, we study a class of posynomial geometric programming problem (PGPF), with the purpose of minimizing a posynomial subject to fuzzy relational equations with max–product composition. With the help of auxiliary variables, it is converted convert the PGPF into an equivalent programming problem whose objective function is a non-decreasing function with an auxiliary variable. Some preliminary definitions are introduced. Since the feasible solutions are not convex the basic programming techniques are not applicable. It is shown that an optimal solution consists of a maximum feasible solution and finite number of minimal feasible solutions by an equivalent programming problem. In fact, there are a lot minimal solutions and to obtain all them is tedious steps, boring and time consuming. Furthermore, we propose some rules for full simplifying the problem. Then by using a branch and bound approach and fuzzy relational equations (FRE) path, it is presented an algorithm to achieve an optimal solution to the PGPF. Finally, a numerical experiment is given to illustrate the steps of the algorithm.

The Aluffi algebra is an algebraic version of characteristic cycles in intersection theory which is an intermediate graded algebra between the symmetric algebra (naive blowup) and the Rees algebra (blowup). Let  R be a commutative Noetherian ring and J ⊂I  ideals of R. We say that J ⊂I  satisfy linearity condition if the Aluffi algebra of I/J is isomorphic with the symmetric algebra. In this paper, we present the necessary and sufficient conditions for satisfying linearity condition. We give classes of ideals J ⊂I   which satisfy linearity condition. We prove that if J is a complete intersection ideal contained in I, then    J ⊂I  satisfy linearity condition if and only if the ideal I is of linear type.

Introduction Many Suppose that is a Banach Space with topological dual space We will denote by  the duality pairing between X and  . For , we denote by the boundary points of Ω and by  the interior of Ω. Also we will denote by  the real nonnegative numbers. Let be a set-valued map from  to . The domain and graph of  are, respectively, defined by We recall that a set valued operator is monotone if   for all  and   For two multivalued operators  and we write  ifis an extension of , i.e., . A monotone operator is called maximal monotone if it has no monotone extension other than itself. In 1988, The Fitzpatrick function of a monotone operator was introduced by Fitzpatrick. The Fitzpatrick function makes a bridge between the results of convex functions and results on maximal monotone operators. For a monotone operator , its Fitzpatrick function is defined byIt is a convex and norm to weak lower semicontinuous and function. Let be an extended real-valued function. Its effective domain is defined by  The function is called proper if . Let  be a proper function. The subdifferential (in the sense of Convex Analysis) of  at  is defined byGiven a proper function  and a map , then  is called -convex ifFor all and for all Given an operator  and a map . Then is called -monotone if for all  and  we have  Also is called maximal -monotone if it has no -monotone extension other than itself. We recall for a proper function  the -subdifferential of  at  is defined by and  if .  Main results   The definition we use for the Fitzpatrick function is the same as for monotone operators. Assume that is a proper -convex function its conjugate is defined by First we have the following refinement of the Fenchel-Moreau inequality: where is the indicator function and . Also we have the following refinement, when is a proper, -convex and lower semicontinuous function and  is a maximal -monotone operator:Moreover, we approach generalized monotonicity from the point of view of the classical concept of polarity. Besides, we introduce and study the notion of generalized monotone polar of a set A. Moreover, we find some equivalent relations between polarity and maximal generalized monotonicity.

A clutter $mathcal{C}$ with vertex set $[n]$ is an antichain of subsets of $[n]$, called circuits, covering all vertices. The clutter is $d$-uniform if all of its circuits have the same cardinality $d$. If $mathbb{K}$ is a field, then there is a one-to-one correspondence between clutters on $V$ and square-free monomial ideals in $mathbb{K}[x_1,ldots,x_n]$ as follows: To each clutter $mathcal{C}$ we correspond its circuit ideal $I(mathcal{C})$ generated by monomials $x_{i_1}cdots x_{i_k}$ with ${i_1,ldots,i_k}inmathcal{C}$. Conversely, to each square-free monomial ideal $I$ with minimal set of generators $mathcal{G}(I)$, we correspond a clutter with circuits ${i_1,ldots,i_k}$, where $x_{i_1}cdots x_{i_k}inmathcal{G}(I)$. The independence complex of a clutter $mathcal{C}$ on $[n]$ is the simplicial complex $Delta_{mathcal{C}}$ whose faces are independent sets in $mathcal{C}$ by which we mean sets $Fsubseteq [n]$ such that $ensubseteq F$ for all $einmathcal{C}$. It is easy to see that the Stanley-Reisner ideal of $Delta_{mathcal{C}}$ coincides with $I(mathcal{C})$. The above correspondence establishes a one-to-one correspondence between simplicial complexes and independence complex of clutters. A simplicial complex $Delta$ is shellable if there exists a total order on its facets, say $F_1<cdots
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In this paper, we solve the multi-period portfolio optimization problem under new assumptions. Recently, the authors examined some distributions instead of Gaussian to fit returns to improve the optimization problem and indicated, by Kolmogorov-Smirnov test, that the Kernel density estimator is the best one. In the present paper, we consider the most appropriate distribution of each asset in each period to fit return. Moreover, in order to optimize the solutions, we use piecewise linear interpolation of the empirical distribution. Furthermore, we show that how one may plan for future investments using historical data and AR(1)-GARCH(1,1) model under new assumptions.

The mann fixed point algorithm play an importmant role in the approximation of fixed points of nonexpansive operators. In this paper, by considering new conditions, we prove the weak convergence of mann fixed point algorithm, for finding a common fixed point of two nonexpansive mappings in real Hilbert spaces. This results extend the privious results given by Kanzow and Shehu. Finally, we give an application of our results, by using the John von Neumann's method.

This paper presents a new stenographic scheme based on matrix embedding using BCH syndrome coding. The proposed method embeds massage into cover by changing some coefficients of cover. In this paper defining a number :::as char:::acteristic of the syndrome, which is invariant with respect to the cyclic shift, we  propose a new embedding algorithm base on BCH  syndrome coding, without finding roots of quadratic and cubic polynomials in Galois field. The computational complexity of the proposed method is linear and space complexity is polynomial of order 2.

Let R be a commutative ring with identity and Nil(R) be the set of nilpotent elements of R. The nil-graph of ideals of R is defined as the graph AG_N(R) whose vertex set is {I:(0)and there exists a non-trivial ideal  such that  and two distinct vertices  and  are adjacent if and only if . Here, we study conditions under which  is complete or bipartite. Also, the independence number of  is determined, where  is a reduced ring. Finally, we classify Artinian rings whose nil-graphs of ideals have genus at most one.

Multivariate time series data, often, modeled using vector autoregressive moving average (VARMA) model. But presence of outliers can violates the stationary assumption and may lead to wrong modeling, biased estimation of parameters and inaccurate prediction. Thus, detection of these points and how to deal properly with them, especially in relation to modeling and parameter estimation of VARMA model is necessary. By detecting outliers, their effect can be eliminated over time and we obtain the modified data. Using this modified data, the proper estimates of the VARMA model are obtained which have the least effect on the outliers. On the other hand, detect of outliers is important in finding an external event over time. For example, by finding outliers in river water monitoring data, flood times can be obtained. The parameter estimation of VAR model is less time consuming than VARMA. On the other hand, under condition of invertibility, VARMA models could be approximated by VAR(p) for large p. Therefore, we use this model to fit and investigate the data generated from VARMA models that contaminated by outliers. Multivariate observations of time series may be contaminated with different types of outliers. However, the effect of different types of outliers in multivariate and univariate case is different, and this observation must be assessed by multivariate approach. In this research, we use a Genetic Algorithm (GA) to develop a procedure for detecting different types of outliers (additive, innovation, level shift and temporary change outliers) in a multivariate time series. GA detects outlier location which minimizes Akaike-like Information Criterion (AIC) and we try to "minimize the number of outliers" and "maximize the likelihood function".  GA is a numerical optimization algorithm whose idea is based on natural selection and natural genetics. This algorithm does not require strong assumptions to obtain the optimal value of a function and has the ability to search for the optimal solution from a space with several local optimal. That is, for example, if a function has several relative maxima, GA finds the absolute maximum of this function as well. For minimization of a function, GA operates by first generating, at random or optionally, several minimal solutions to the function that this set of solutions called the initial population and each solution as a chromosome. Then, using reproductive operators, we combine chromosomes and make a jump into them. If the function of newly produced chromosomes is lower than the previous chromosomes, these chromosomes can be added to the initial population or replaced with chromosomes with less function in this population. This process is repeated until convergence occurs or the end number of itteration obtained. Furthermore, we introduce another method of detecting outliers, the Tsay Pena and Pankratz (TPP) method. TPP uses some test statistics based on outliers size and VAR parameters. This method detects outliers in three stages. In stage I, it detects one by one outliers and remove their effects. Iteration done until no outlier found. In stage II, for detected outlier in stage I, the estimation of outliers effects are obtained simultaneously. Then, outliers with insignificant effects are removed. The VAR parameters re-estimated based on modified series of this stage. In stage III, we repeated stage I and II with new VAR parameter estimation. In each iteration of TPP, an outlier is detected and the effect of this outlier is removed from series (modified series). Then the parameter estimation is obtained from the modified series and the next outlier detection is continued using these estimates. This may lead to biased estimates and wrong detection of the next outlier point. In other words, in the TPP method, one detected outlier hides another outlier (masking), or one detected outlier reveals the usual observation as an outlier (swamping). This method often mis-detects the type of outliers. But in each iteration of GA, a random pattern of outliers (for testing) is first generated and a temporary modified series is obtained by removing effect of this pattern from series. Then the estimation of the parameters obtained and the detection of this pattern is tested. This work reduces the effect of the previously identified outliers on the full pattern of the outliers. In fact, if the random pattern of all outliers is correctly generated, almost effect of all of them will be eliminated in the modified series. Therefore using this temporary modified series, the GA obtained more accurate estimates and detected outliers more accurately. The simulation results confirm the validity of the GA method and the percentage of correct outlier detection in this method is higher than the TPP method. GA, of course, needs more time to calculate. Also, although the VAR model is used in both detection methods, the percentage of correct outlier detection in the VARMA model data is similar to the VAR model. Gas-furnace data were analyzed and modeled and it was determined that GA and TPP methods detected similar outliers. Fitting the VAR(6) model on these data shows that the variance of input gas error in modified data of GA to TPP is reduced by 17% and the variance of carbon dioxide error in the modified data of GA to TPP reduced by 43%.