In this paper, the dynamical behavior of a three-dimensional fractional-order prey-predator model is investigated with Holling type III functional response and constant rate harvesting. It is assumed that the middle predator species consumes only the prey species, and the top predator species consumes only the middle predator species. We also prove the boundedness, the non-negativity, the uniqueness, and the existence of the solutions of the proposed model. Then, all possible equilibria are determined, and the dynamical behaviors of the proposed model around the equilibrium points are investigated. Finally, numerical simulations results are presented to confirm the theoretical results and to give a better understanding of the dynamics of our proposed model.

This paper proposes a new fitted operator strategy for solving singularly perturbed parabolic partial differential equation with delay on the spatial variable. We decomposed the problem into three piecewise equations. The delay term in the equation is expanded by Taylor series, the time variable is discretized by implicit Euler method, and the space variable is discretized by central difference methods. After developing the fitting operator method, we accelerate the order of convergence of the time direction using Richardson extrapolation scheme and obtained $O\left(h^2+k^2\right)$ uniform order of convergence. Finally, three examples are given to illustrate the effectiveness of the method. The result shows the proposed method is more accurate than some of the methods that exist in the literature.

In this paper, we consider a source problem for a time harmonic acoustic wave in two-dimensional space. Based on the boundary integral equation method, a Dirichlet-to-Neumann map in terms of boundary integral operators on the boundary of the source is constructed to transform this problem into two boundary value problems for the Helmholtz equation.

A method for the group classification of differential equations is proposed. It is based on the determination of all possible cases of linear dependence of certain indeterminates appearing in the determining equations of symmetries of the equation. The method is simple and systematic and applied to a family of hyperbolic equations. Moreover, as the given family contains several known equations with important physical applications, low-order conservation laws of some relevant equations from the family are computed, and the results obtained are discussed with regard to the symmetry integrability of a particular class from the underlying family of hyperbolic equations.

In this paper, we introduce original definitions of Partner ruled surfaces according to the Darboux frame of a curve lying on an arbitrary regular surface in $E^3$. It concerns $Tg$ Partner ruled surfaces, $Tn$ Partner ruled surfaces, and $gn$ Partner ruled surfaces. We aim to study the simultaneous developability conditions of each couple of two Partner ruled surfaces. Finally, we give an illustrative example for our study.

In this article, we will recall the main properties of the Fourier transform on the Heisenberg motion group $G={ℍ}^n⋊K$, where $K=U\left(n\right)$ and ${ℍ}^n={ℂ}^n\times ℝ$ denote the Heisenberg group. Then, we will present some uncertainty principles associated to this transform as Beurling, Hardy, and Gelfand-Shilov.

In this article, we study the existence of solutions for nonlocal $p\left(x\right)$-biharmonic Kirchhoff-type problem with Navier boundary conditions. By different variational methods, we determine intervals of parameters for which this problem admits at least one nontrivial solution.

This paper presents the study of singularly perturbed differential-difference equations of delay and advance parameters. The proposed numerical scheme is a fitted fourth-order finite difference approximation for the singularly perturbed differential equations at the nodal points and obtained a tridiagonal scheme. This is significant because the proposed method is applicable for the perturbation parameter which is less than the mesh size, where most numerical methods fail to give good results. Moreover, the work can also help to introduce the technique of establishing and making analysis for the stability and convergence of the proposed numerical method, which is the crucial part of the numerical analysis. Maximum absolute errors range from ${10}^{-03}$ up to ${10}^{-10}$, and computational rate of convergence for different values of perturbation parameter, delay and advance parameters, and mesh sizes are tabulated for the considered numerical examples. Concisely, the present method is stable and convergent and gives more accurate results than some existing numerical methods reported in the literature.

In this article, we propose two Banach-type fixed point theorems on bipolar metric spaces. More specifically, we look at covariant maps between bipolar metric spaces and consider iterates of the map involved. We also propose a generalization of the Banach fixed point result via Caristi-type arguments.

The Jacobi elliptic function method is applied to solve the generalized Benjamin-Bona-Mahony equation (BBM). Periodic and soliton solutions are formally derived in a general form. Some particular cases are considered. A power series method is also applied in some particular cases. Some solutions are expressed in terms of the Weierstrass elliptic function.

Variational methods are used in order to establish the existence of nontrivial weak solution for superlinear second-order system with noninstantaneous impulses. The main result is obtained when a kind of definition of the weak solution for this system is introduced. Meanwhile, an example is presented to illustrate the main result.

Let $A$ and $B$ be two closed linear relations acting between two Banach spaces $X$ and $Y$, and let $\lambda$ be a complex number. We study the stability of the nullity and deficiency of $A$ when it is perturbed by $\lambda B$. In particular, we show the existence of a constant for which both the nullity and deficiency of $A$ remain stable under perturbation by $\lambda B$ for all $\lambda$ inside the disk .

In this paper, we introduce original definitions of Smarandache ruled surfaces according to Frenet-Serret frame of a curve in $E^3$. It concerns TN-Smarandache ruled surface, TB-Smarandache ruled surface, and NB-Smarandache ruled surface. We investigate theorems that give necessary and sufficient conditions for those special ruled surfaces to be developable and minimal. Furthermore, we present examples with illustrations.

This paper uses the generalization of the Hukuhara difference for compact convex set to extend the classical notions of Carathéodory differentiability to multifunctions (set-valued maps). Using the Hukuhara difference and affine multifunctions as a local approximation, we introduce the notion of CH-differentiability for multifunctions. Finally, we tackle the study of the relation among the Fréchet differentiability, Hukuhara differentiability, and CH-differentiability.

We introduce and study two properties of dynamical systems: topologically transitive and topologically mixing under the set-valued setting. We prove some implications of these two properties for set-valued functions and generalize some results from a single-valued case to a set-valued case. We also show that both properties of set-valued dynamical systems are equivalence for any compact intervals.

Frame theory has a great revolution for recent years. This theory has been extended from Hilbert spaces to Hilbert $C^{\ast }$-modules. In this paper, we define and study the new concept of controlled continuous $\ast$-$K$-$g$-frames for Hilbert $C^{\ast }$-modules and we establish some properties.

The main goal of this paper is to investigate the boundedness and essential norm of a class of product-type operators $\left(T_{u,v,\phi }^m,m\in \mathbb{N}\right)$ from Hardy spaces into $n$th weighted-type spaces. As a corollary, we obtain some equivalent conditions for compactness of such operators.

This paper presents the study of singularly perturbed differential equations of convection-diffusion type with nonlocal boundary condition. The proposed numerical scheme is a combination of the classical finite difference method for the boundary conditions and exponential fitted operator method for the differential equations at the interior points. Maximum absolute errors and rates of convergence for different values of perturbation parameter and mesh sizes are tabulated for the numerical examples considered. The method is shown to be first-order accuracy independent of the perturbation parameter $\epsilon$.

For the superreplication problem with discrete time, a guaranteed deterministic formulation is considered: the problem is to guarantee coverage of the contingent liability on sold option under all admissible scenarios. These scenarios are defined by means of a priori defined compacts dependent on price prehistory: the price increments at each point in time must lie in the corresponding compacts. In a general case, we consider a market with trading constraints and assume the absence of transaction costs. The formulation of the problem is game theoretic and leads to the Bellman–Isaacs equations. This paper analyses the solution to these equations for a specific pricing problem, i.e., for a binary option of the European type, within a multiplicative market model, with no trading constraints. A number of solution properties and an algorithm for the numerical solution of the Bellman equations are derived. The interest in this problem, from a mathematical prospective, is related to the discontinuity of the option payoff function.

This article deals with some existence, uniqueness, and Ulam-Hyers-Rassias stability results for a class of boundary value problem for nonlinear implicit fractional differential equations with impulses and generalized Hilfer Fractional derivative. The results are obtained using the Banach contraction principle and Krasnoselskii’s and Schaefer’s fixed-point theorems.

In this paper, we introduce an almost generalized $\alpha$-admissible $Z$-contraction with the help of a simulation function and study fixed point results in the setting of partially ordered b-metric spaces. The presented results generalize and unify several related fixed point results in the existing literature. Finally, we verify our results by using two examples. Moreover, one of our fixed point results is applied to guarantee the existence of a solution of an integral equation.

In this paper, the notion of set-valued $\mathcal{L}$-contractions is introduced, and a new fixed point theorem for such contractions is established. An example to illustrate main theorem is given.

This paper is aimed at proving a common fixed point theorem for $F$-Kannan mappings in metric spaces with an application to integral equations. The main result of the paper will extend and generalise the recent existing fixed point results in the literature. We also provided illustrative examples and some applications to integral equation, nonlinear fractional differential equation, and ordinary differential equation for damped forced oscillations to support the results.

The primary purpose of this research is to demonstrate an efficient replacement of double transform called the double Laplace–Sumudu transform (DLST) and prove some related theorems of the new double transform. Also, we will discuss the fundamental properties of the double Laplace–Sumudu transform of some basic functions. Then, by utilizing those outcomes, we will apply it to the partial differential equations to show its simplicity, efficiency, and high accuracy.

This paper introduces novel concepts of joint $\left(Y,Z\right)$ cyclic $G‐{\Omega }_{S,T,\left(abef\right)}$-weak contraction and joint $\left(Y,Z\right)$ cyclic $G‐{\Omega }_{S,T,\left(abef\right)}$-weak nonexpansive mappings and then proves the existence of a unique common fixed point of such mappings in case of complete and compact metric spaces, respectively, in particular, it proves the existence of a unique fixed point for both cyclic $G‐{\Omega }_{S,\left(abef\right)}$-weak contraction and cyclic $G‐{\Omega }_{S,\left(abef\right)}$-weak nonexpansive mappings, and hence, it also proves the existence of a unique fixed point for both cyclic ${\Omega }_{S,\left(abef\right)}$-weak contraction and cyclic ${\Omega }_{S,\left(abef\right)}$-weak nonexpansive mappings. The results of this research paper extend and generalize some fixed point theorems previously proved via the attached references.

In this paper, we present the notion of $\theta -\varphi -$ expansive mapping in complete rectangular metric spaces and study various fixed point theorems for such mappings. The presented theorems extend, generalize, and improve many existing results in the literature.

The nonlinear wave equation is a significant concern to describe wave behavior and structures. Various mathematical models related to the wave phenomenon have been introduced and extensively being studied due to the complexity of wave behaviors. In the present work, a mathematical model to obtain the solution of the nonlinear wave by coupling the classical Camassa-Holm equation and the Rosenau-RLW-Kawahara equation with the dual term of nonlinearities is proposed. The solution properties are analytically derived. The new model still satisfies the fundamental energy conservative property as the original models. We then apply the energy method to prove the well-posedness of the model under the solitary wave hypothesis. Some categories of exact solitary wave solutions of the model are described by using the Ansatz method. In addition, we found that the dual term of nonlinearity is essential to obtain the class of analytic solution. Besides, we provide some graphical representations to illustrate the behavior of the traveling wave solutions.

This paper introduces a new class of mappings called $S$-Geraphty-contractions and provides sufficient conditions for the existence and uniqueness of a best proximity point for such mappings. It also presents the best proximity point result for generalized contractions as well. Our results extend and generalize some theorems in the literature.

This paper is devoted to study the null controllability properties of a nonlinear age and two-sex population dynamics structured model without spatial structure. Here, the nonlinearity and the couplage are at the birth level. In this work, we consider two cases of null controllability problem. The first problem is related to the extinction of male and female subpopulation density. The second case concerns the null controllability of male or female subpopulation individuals. In both cases, if $A$ is the maximal age, a time interval of duration $A$ after the extinction of males or females, one must get the total extinction of the population. Our method uses first an observability inequality related to the adjoint of an auxiliary system, a null controllability of the linear auxiliary system, and after Kakutani’s fixed-point theorem.

In this research paper, we have set some related fixed point results for generalized weakly contractive mappings defined in partially ordered complete $b$-metric spaces. Our results are an extension of previous authors who have already worked on fixed point theory in $b$-metric spaces. We state some examples and one sample of the application of the obtained results in integral equations, which support our results.

This paper intends to obtain accurate and convergent numerical solutions of linear space-time matching telegraph fractional equations by means of a double Sumudu matching transformation method. Moreover, the numerical model is equipped to explain the work, the accuracy of the work, and sobriety in its presentation method, and as a result, the proposed method shows an effective and convenient way, to employ proven problems in science and engineering.

In this paper, I consider the Cauchy problem for the incompressible Navier-Stokes equations in ${ℝ}_{+}^n$ for $n\ge 3$ with bounded initial data and derive a priori estimates of the maximum norm of all derivatives of the solution in terms of the maximum norm of the initial data. This paper is a continuation of my work in my previous papers, where the initial data are considered in ${𝕋}^n$ and ${ℝ}^n$ respectively. In this paper, because of the nonempty boundary in our domain of interest, the details in obtaining the desired result are significantly different and more challenging than the work of my previous papers. This challenges arise due to the possible noncommutativity nature of the Leray projector with the derivatives in the direction of normal to the boundary of the domain of interest. Therefore, we only consider one derivative of the velocity field in that direction.

In this paper, we give an equivalent characterization of the Besov space. This reveals the equivalent relation between the mixed derivative norm and single-variable norm. Fourier multiplier, real interpolation, and Littlewood-Paley decomposition are applied.

The aim of this paper is to extend the notion of $𝕂$-Riemann integrability of functions defined over $\left[a,b\right]$ to functions defined over a rectangular box of ${ℝ}^n$. As a generalization of step functions, we introduce a notion of $𝕂$-step functions which allows us to give an equivalent definition of the $𝕂$-Riemann integrable functions.

The ultimate objective of the problem under study is to apply the min-max tool, thus making it possible to optimize the default risks linked to several areas: the agricultural sector, for example, which requires the optimization of the default risk using the following elements: silage crops, annual consumption requirements, and crops produced for a given year. To minimize the default risk in the future, we start, in the first step, by forecasting the total budget of agriculture investment for the next 20 years, then distribute this budget efficiently between the irrigation and construction of silos. To do this, Bangladesh was chosen as an empirical case study given the availability of its data on the FAO website; it is considered a large agricultural country in South Asia. In this article, we give a detailed and original in-depth study of the agricultural planning model through a calculating algorithm suggested to be coded on the R software thereafter. Our approach is based on an original statistical modeling using nonparametric statistics and considering an example of a simulation involving agricultural data from the country of Bangladesh. We also consider a new pollution model, which leads to a vector optimization problem. Graphs illustrate our quantitative analysis.

We prove lower semicontinuity in $L^1\left(\Omega \right)$ for a class of functionals $G:BV\left(\Omega \right)⟶ℝ$ of the form $G\left(u\right)={\int }_{\Omega }g\left(x,\nabla u\right)dx+{\int }_{\Omega }\psi \left(x\right)d\left|D^{s}u\right|$ where $g:\Omega \times {ℝ}^N⟶ℝ$, $\Omega \subset {ℝ}^N$ is open and bounded, $g\left(·,p\right)\in L^1\left(\Omega \right)$ for each $p,$ satisfies the linear growth condition $\underset{\left|p\right|⟶\infty }{\lim }g\left(x,p\right)/\left|p\right|=\psi \left(x\right)\in C\left(\Omega \right)\cap L^{\infty }\left(\Omega \right),$ and is convex in $p$ depending only on $\left|p\right|$ for a.e. $x.$ Here, we recall for $u\in BV\left(\Omega \right)$; the gradient measure $Du=\nabla udx+d\left(D^{s}u\right)\left(x\right)$ is decomposed into mutually singular measures $\nabla udx$ and $d\left(D^{s}u\right)\left(x\right)$. As an example, we use this to prove that ${\int }_{\Omega }\psi \left(x\right)\sqrt{{\alpha }^2\left(x\right)+{\left|\nabla u\right|}^2}dx+{\int }_{\Omega }\psi \left(x\right)d\left|D^{s}u\right|$ is lower semicontinuous in $L^1\left(\Omega \right)$ for any bounded continuous $\psi$ and any $\alpha \in L^1\left(\Omega \right).$ Under minor addtional assumptions on $g$, we then have the existence of minimizers of functionals to variational problems of the form $G\left(u\right)+{\left|\left|u-u_0\right|\right|}_{L^1}$ for the given $u_0\in L^1\left(\Omega \right),$ due to the compactness of $BV\left(\Omega \right)$ in $L^1\left(\Omega \right).$

Chaundy and Jolliffe proved that if $\left\{a_n\right\}$ is a nonnegative, nonincreasing real sequence, then series $\sum a_n\sin \left(nx\right)$ converges uniformly if and only if $n{a}_n⟶0$. The purpose of this paper is to show that if $\left\{n{a}_n\right\}$ is nonincreasing and $n{a}_n⟶0$, then the series $f\left(x\right)=\sum a_n\sin \left(\sqrt{n}x\right)$ can be differentiated term-by-term on $\left[c,d\right]$ for . However, $f^{\prime }\left(0\right)$ may not exist.

In this paper, a singularly perturbed second-order ordinary differential equation with discontinuous source term subject to mixed-type boundary conditions is considered. A fitted nonpolynomial spline method is suggested. The stability and parameter uniform convergence of the proposed method are proved. To validate the applicability of the scheme, two model problems are considered for numerical experimentation and solved for different values of the perturbation parameter, $\epsilon$, and mesh size, $h.$ The numerical results are tabulated in terms of maximum absolute errors and rate of convergence, and it is observed that the present method is more accurate and $\epsilon$-uniformly convergent for $h\ge \epsilon$ where the classical numerical methods fail to give good result and it also improves the results of the methods existing in the literature.

The investigation of the numerical solution of the laminar boundary layer flow along with a moving cylinder with heat generation, thermal radiation, and surface slip effect is carried out. The fluid mathematical model developed from the Navier-Stokes equations resulted in a system of partial differential equations which were then solved by the multidomain bivariate spectral quasilinearization method (MD-BSQLM). The results show that increasing the velocity slip factor results in an enhanced increase in velocity and temperature profiles. Increasing the heat generation parameter increases temperature profiles; increasing the radiation parameter and the Eckert numbers both increase the temperature profiles. The concentration profiles decrease with increasing radial coordinate. Increasing the Brownian motion and the thermophoresis parameter both destabilizes the concentration profiles. Increasing the Schmidt number reduces temperature profiles. The effect of increasing selected parameters: the velocity slip, Brownian motion, and the radiation parameter on all residual errors show that these errors do not deteriorate. This shows that the MD-BSQLM is very accurate and robust. The method was compared with similar results in the literature and was found to be in excellent agreement.

In this paper, weighted norm inequalities for multilinear Fourier multipliers satisfying Sobolev regularity with mixed norm are discussed. Our result can be understood as a generalization of the result by Fujita and Tomita by using the $L^r$-based Sobolev space, with mixed norm.

This paper presents generalized refinement of Gauss-Seidel method of solving system of linear equations by considering consistently ordered 2-cyclic matrices. Consistently ordered 2-cyclic matrices are obtained while finite difference method is applied to solve differential equation. Suitable theorems are introduced to verify the convergence of this proposed method. To observe the effectiveness of this method, few numerical examples are given. The study points out that, using the generalized refinement of Gauss-Seidel method, we obtain a solution of a problem with a minimum number of iteration and obtain a greater rate of convergence than other previous methods.

In this work, we consider the Darboux frame $\left(T,V,U\right)$ of a curve lying on an arbitrary regular surface and we construct ruled surfaces having a base curve which is a $V$-direction curve. Subsequently, a detailed study of these surfaces is made in the case where the directing vector of their generatrices is a vector of the Darboux frame, a Darboux vector field. Finally, we give some examples for special curves such as the asymptotic line, geodesic curve, and principal line, with illustrations of the different cases studied.

We discuss martingale transforms between martingale Hardy-amalgam spaces $H_{p,q}^s,Q_{p,q}$ and $P_{p,q}.$ Let and and let $f=\left(f_n,n\in \mathbb{N}\right)$ be a martingale in $P_{p_1,q_1}$; then, we show that its martingale transforms are the martingales in $P_{p,q}$ for some $p,q$ and similarly for $H_{p,q}^s$ and $Q_{p,q}.$

In this article, a numerical technique based on the Chebyshev cardinal functions (CCFs) and the Lagrange multiplier technique for the numerical approximation of the variable-order fractional integrodifferential equations are shown. The variable-order fractional derivative is considered in the sense of regularized Hilfer-Prabhakar and Hilfer-Prabhakar fractional derivatives. To solve the problem, first, we obtain the operational matrix of the regularized Hilfer-Prabhakar and Hilfer-Prabhakar fractional derivatives of CCFs. Then, this matrix and collocation method are used to reduce the solution of the nonlinear coupled variable-order fractional integrodifferential equations to a system of algebraic equations which is technically simpler for handling. Convergence and error analysis are examined. Finally, some examples are given to test the proposed numerical method to illustrate its accuracy and efficiency.

In this article, a numerical method based on the shifted Chebyshev functions for the numerical approximation of the coupled nonlinear variable-order fractional sine-Gordon equations is shown. The variable-order fractional derivative is considered in the sense of Caputo-Prabhakar. To solve the problem, first, we obtain the operational matrix of the Caputo-Prabhakar fractional derivative of shifted Chebyshev polynomials. Then, this matrix and collocation method are used to reduce the solution of the nonlinear coupled variable-order fractional sine-Gordon equations to a system of algebraic equations which is technically simpler for handling. Convergence and error analysis are examined. Finally, some examples are given to test the proposed numerical method to illustrate the accuracy and efficiency of the proposed method.

We give different types of new characterizations for the boundedness and essential norms of generalized weighted composition operators between Zygmund-type spaces. Consequently, we obtain new characterizations for the compactness of such operators.

In this article, a numerical solution is proposed for singularly perturbed delay parabolic reaction-diffusion problem with mixed-type boundary conditions. The problem is discretized by the implicit Euler method on uniform mesh in time and extended cubic B-spline collocation method on a Shishkin mesh in space. The parameter-uniform convergence of the method is given, and it is shown to be $\epsilon$-uniformly convergent of $O\left(\Delta t+N^{-2}{\ln }^{2}N\right)$, where $\Delta t$ and $N$ denote the step size in time and number of mesh intervals in space, respectively. The proposed method gives accurate results by choosing suitable value of the free parameter $\lambda$. Some numerical results are carried out to support the theory.

In this work, the existence of at least one solution for the following third-order integral and $m$-point boundary value problem on the half-line at resonance ${\left(\rho \left(t\right)u^{\text{'}}\left(t\right)\right)}^{\text{'}\text{'}}=w\left(t,u\left(t\right),u^{\text{'}}\left(t\right),u^{\text{'}\text{'}}\left(t\right)\right),t\in [0,\infty ),u\left(0\right)={\sum }_{j=1}^m{\alpha }_j{\int }_0^{{\eta }_j}u\left(t\right)dt,u^{\text{'}}\left(0\right)=0,\underset{t⟶\infty }{\lim }{\left(\rho \left(t\right)u^{\text{'}}\left(t\right)\right)}^{\text{'}}=0,$ will be investigated. The Mawhin’s coincidence degree theory will be used to obtain existence results while an example will be used to validate the result obatined.

In this work, we develop an a posteriori error analysis of a conforming mixed finite element method for solving the coupled problem arising in the interaction between a free fluid and a fluid in a poroelastic medium on isotropic meshes in ${ℝ}^d$, $d\in \left\{2,3\right\}$. The approach utilizes a Lagrange multiplier method to impose weakly the interface conditions [Ilona Ambartsumyan et al., Numerische Mathematik, 140 (2): 513-553, 2018]. The a posteriori error estimate is based on a suitable evaluation on the residual of the finite element solution. It is proven that the a posteriori error estimate provided in this paper is both reliable and efficient. The proof of reliability makes use of suitable auxiliary problems, diverse continuous inf-sup conditions satisfied by the bilinear forms involved, Helmholtz decomposition, and local approximation properties of the Clément interpolant. On the other hand, inverse inequalities and the localization technique based on simplexe-bubble and face-bubble functions are the main tools for proving the efficiency of the estimator. Up to minor modifications, our analysis can be extended to other finite element subspaces yielding a stable Galerkin scheme.

In this article, small modification to the Modified Euler Method is proposed. Stability and consistency were tested to determine the end result, and some numerical results were presented, and the CPU time was compared again, and it is recognized that the proposed method is more reliable and compatible with higher efficiency.

We present the concept of $\left(\alpha ,k,\theta ,\phi \right)$-contractive multivalued mappings in $b$-metric spaces and prove some fixed point results for these mappings in this study. Our results expand and refine some of the literature’s findings in fixed point theory.

In the present paper, we establish some composition formulas for Marichev-Saigo-Maeda (MSM) fractional calculus operators with $V$-function as the kernel. In addition, on account of $V$-function, a variety of known results associated with special functions such as the Mittag-Leffler function, exponential function, Struve’s function, Lommel’s function, the Bessel function, Wright’s generalized Bessel function, and the generalized hypergeometric function have been discovered by defining suitable values for the parameters.