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The present paper extends some previous works studying automorphisms in type-2 fuzzy sets. The framework for the analysis is the set of convex and normal functions from [ 0 , 1 ] to [ 0 , 1 ] (fuzzy truth values). The paper concentrates on those automorphisms that, in this framework, leave the constant function 1 fixed. This function is quite important since it defines the boundary between the functions that represent “TRUE” (increasing functions) and those that represent “FALSE” (decreasing functions), being at the same time the only normal function that is simultaneously increasing and decreasing. While C.L. Walker, E.A. Walker and J. Harding introduced in 2008 a family of functions leaving the constant function 1 fixed, the main goal of this paper is to prove that the functions of that family are in fact automorphisms, and moreover, that they are the only automorphisms (in the mentioned set of convex and normal functions from [ 0 , 1 ] to [ 0 , 1 ] ) that preserve the function 1.

We study the optimal bids and allocations in a real-time auction for heterogeneous items subject to the requirement that specified collections of items of given types be acquired within given time constraints. The problem is cast as a continuous time optimization problem that can, under certain weak assumptions, be reduced to a convex optimization problem. Focusing on the standard first and second price auctions, we first show, using convex duality, that the optimal (infinite dimensional) bidding policy can be represented by a single finite vector of so-called ''pseudo-bids''. Using this result we are able to show that the optimal solution in the second price case turns out to be a very simple piecewise constant function of time. This contrasts with the first price case that is more complicated. Despite the fact that the optimal solution for the first price auction is genuinely dynamic, we show that there remains a close connection between the two cases and that, empirically, there is almost no difference between optimal behavior in either setting. This suggests that it is adequate to bid in a first price auction as if it were in fact second price. Finally, we detail methods for implementing our bidding policies in practice with further numerical simulations illustrating the performance.

In this study, we investigate the existence of a solution to the boundary value problem (BVP) of variable-order Caputo-type fractional differential equation by converting it into an equivalent standard Caputo (BVP) of the fractional constant order with the help of the generalized intervals and the piecewise constant functions. All our results in this study are proved by using Darbo’s fixed-point theorem and the Ulam–Hyers (UH) stability definition. A numerical example is given at the end to support and validate the potentiality of our obtained results.

Ethnomathematics can empirically improve the cognitive abilities of students in elementary and secondary schools. However, in undergraduate study, there are still limited studies on integrating ethnomathematics in learning resources. This study aims to apply interpolation in modelling polynomial functions and integral volume on the shape of Buginese and Makassarese traditional foods. Furthermore, it can be used by students as relevant learning resources regarding interpolation and the concept of volume of solid of revolution (VOSR). This is a qualitative study using an ethnographic approach. The data were collected through observations to obtain general information, interviews with informants to find out food-making techniques, and documentation to obtain physical models of each type of food. Data Analysis Techniques consist of the domain analysis to obtain an overview of Buginese and Makassarese traditional foods and the taxonomic analysis to categorize mathematical concepts obtained from the modeling and simulation. The result of this research reveals that lammang is suitable with the slabs. It can be represented as constant functions that revolved around the x -axis or the y -axis. While paso , bolu cukke, and cantik manis as well as barongko batara , Putu, and cucuru can be outlined in linear functions rotating about the x -axis, y -axis, or others fixed-line. They meet the criteria of the disks method. However, they are described in the function of polynomials of n-degree. The use of washers can be described in the model of blundered and sarang semut with a hole in the middle caused by the intersection of two curves rotated about the x -axis or the y -axis. For shells, the model can be applied to determine the cover volume of the cover of pisang ijo flour and onde-onde. Thus, all types of traditional foods in this study can be appropriate objects for a learning resource in modelling the VOSR.

Motivated by the trade-off issue between delay performance and energy consumption in modern computer and communication systems, we consider a single-server queue with phase-type service requirements and with the following two special features: Firstly, the service speed is a piecewise constant function of the workload. Secondly, the server switches off when the system becomes empty, only to be activated again when the workload reaches a certain threshold. For this system, we obtain the steady-state workload distribution and its moments of any order. We use this result to choose the activation threshold such that a certain cost function, involving processing costs, activation costs and mean workload, is minimized.

In this work it is considered the Fick’s second law in a context in which the diffusion coefficient depends on the concentration. It is employed the Kirchhoff transformation in order to simplify the mathematical structure of the Fick’s second law, giving rise to a more convenient description. In order to provide a general protocol, the diffusion coefficient will be assumed a piecewise constant function of the concentration. Exact formulas are presented for both the Kirchhoff transformation and its inverse, in such a way that there is no limit of accuracy. Some numerical examples are presented with the aid of a semi-implicit procedure associated with a finite difference approximation.

We consider a two-level open quantum system whose dynamics is governed by the Gorini–Kossakowski–Sudarshan–Lindblad equation with Hamiltonian and dissipation superoperator depending, respectively, on coherent and incoherent controls. Results about reachability, controllability, and minimum-time control are obtained in terms of the Bloch parametrization. First, we consider the case when the zero coherent and incoherent controls satisfy the Pontryagin maximum principle in the class of piecewise continuous controls. Second, for zero coherent control and for incoherent control lying in the class of constant functions, the reachability and controllability sets of the system are exactly described and some analytical results on the minimum-time control are found. Third, we consider a series of increasing values of the final time and the corresponding classes of controls with zero incoherent control and with coherent control equal to zero until a switching time instant and to a cosine function after it. The corresponding reachable points in the Bloch ball are numerically obtained and visualized. Fourth, a known method for estimating reachable sets is adapted and used to analyze the situation where the zero coherent and incoherent controls satisfy the Pontryagin maximum principle in the class of piecewise continuous controls while, as shown numerically, are not optimal.

Before going further with fractional derivative which is constructed by Rabotnov exponential kernel, there exist many questions that are not addressed. In this paper, we try to recapitulate all the fundamental calculus, which we can obtain with this new fractional operator. The problems in this paper are to determine the solutions of the fractional differential equations where the second members are constant functions, polynomial functions, exponential functions, trigonometric functions, or Mittag-Leffler functions. For all the fractional differential equations, the obtained solutions are represented graphically. The Laplace transform of the fractional derivative with Rabotnov exponential kernel is the primary tool in the investigations. Finally, we give the fundamental solution to the nonlinear time-fractional modified Degasperis–Procesi equation by considering the fractional operator with Rabotnov exponential kernel.

This paper investigates total variation minimization in one spatial dimension for the recovery of gradient-sparse signals from undersampled Gaussian measurements. Recently established bounds for the required sampling rate state that uniform recovery of all $s$-gradient-sparse signals in ${\mathbb{R}}^n$ is only possible with $m \gtrsim \sqrt{s n} \cdot{\operatorname{PolyLog}}(n)$ measurements. Such a condition is especially prohibitive for high-dimensional problems, where $s$ is much smaller than $n$. However, previous empirical findings seem to indicate that this sampling rate does not reflect the typical behavior of total variation minimization. The present work provides a rigorous analysis that breaks the $\sqrt{s n}$-bottleneck for a large class of “natural” signals. The main result shows that non-uniform recovery succeeds with high probability for $m \gtrsim s \cdot{\operatorname{PolyLog}}(n)$ measurements if the jump discontinuities of the signal vector are sufficiently well separated. In particular, this guarantee allows for signals arising from a discretization of piecewise constant functions defined on an interval. The key ingredient of the proof is a novel upper bound for the associated conic Gaussian mean width, which is based on a signal-dependent, non-dyadic Haar wavelet transform. Furthermore, a natural extension to stable and robust recovery is addressed.

In this paper, a new time-space network model is proposed for addressing the time-dependent rural postman problem (TDRPP) of a single vehicle. The proposed model follows the idea of arc-path alternation to form a feasible and complete route. Based on the proposed model, the time dependency of the TDRPP is better described to capture its dynamic process, compared to the existing methods using a piecewise constant function with limited intervals. Furthermore, the property of first-in-first-out (FIFO) can be satisfied with the time spent on each arc. We investigate the FIFO property for the considered time-dependent network and key optimality property for the TDRPP. Based on this property, a dedicated genetic algorithm (GA) is proposed to efficiently solve the considered TDRPP that suffers from computational intractability for large-scale cases. Comprehensive simulation experiments are conducted for various time-dependent networks to show the effectiveness of the proposed GA.

Identities that relate projections of Raviart–Thomas finite element vector fields to discrete gradients of Crouzeix–Raviart finite element functions are derived under general conditions. Various implications such as discrete convex duality results and a characterization of the image of the projection of the Crouzeix–Ravaiart space onto elementwise constant functions are deduced.

In this paper, we introduce a nonlinear Lanchester-type model involving supply units. The model describes a battle where the Blue party consisting of one armed force $$B $$ is fighting against the Red party. The Red party consists of $$n $$ armed forces each of which is supplied by a supply unit. A new variable called “fire allocation” is associated to the Blue force, reflecting its strategy during the battle. A problem of optimal fire allocation for Blue force is then studied. The optimal fire allocation of the Blue force allows that the number of Blue troops is always at its maximum. The allocation is sought in the form of a piece-wise constant function of time with the help of “threatening rates” computed for each agent of the Red party. Numerical experiments are included to justify the theoretical results.

The key goal of this article is to propose a modification of certain exponential type operators defined by Ismail and May. Particularly, we concentrate on a sequence of operators that preserve $$e^{-x}$$ and constant functions. We find the moments of these modified operators using the concept of moment generating function with the help of Mathematica software. We show uniform convergence of these modified operators and analyze the asymptotic behaviour with a Voronovskaya type theorem. We also illustrate via graphs that our modified operators approximate better than the original operators for certain family of functions. Finally, we show the convergence of these modified operators graphically using Mathematica Software.

We investigate the spectral properties of discrete one-dimensional Schrodinger operators whose potentials are generated by sampling along the elements of the Fibonacci subshift with a locally constant function. The fundamental trace map formalism for this model is presented and related to its spectral features via an extension of a multitude of works on the classical model, where the sampling function only depends on a single entry of the sequence.

In this paper, we discuss a priori error estimates and superconvergence of $$P_0^2$$ – $$P_1$$ mixed finite element methods for elliptic boundary control problems. The state variables and co-state variables are approximated by the $$P_0^2$$ – $$P_1$$ (velocity-pressure) pair, and the control variable is approximated by piecewise constant functions. First, we derive a priori error estimates for the control variable, the state variables, and the co-state variables. Then we obtain a superconvergence result for the control variable by using a postprocessing projection operator.

This paper investigates a production inventory model under classical EPQ framework with the assumption that the customer demand during the stock-out period is affected by the accumulated back-orders. The backlog rate is not fixed; instead, the demand during stock-out is assumed to decrease proportionally to the existing backlog which is thereby approximated by a piecewise constant function. Deteriorating items are taken into consideration in this work. For better illustration of the theoretical results and to highlight managerial insights, numerical examples are presented which are then compared to the results obtained by considering an exact (non-approximated) backlogging rate (from literature). The comparisons indicate high quality results for the approximated model.

In contemporary mathematics, parameterization tool like soft set theory precisely tackle complex problems of economics and engineering. In this paper, we demonstrate a novel approach of soft set theory i.e., intersectional soft (int-soft) sets of an ordered $ \Gamma $ -semigroup $ S $ and develop int-soft left (resp. right) $ \Gamma $-ideals of $ S. $ Various classes like $ \Gamma $-regular, left $ \Gamma $-simple, right $ \Gamma $-simple and some semilattices of an ordered $ \Gamma $-semigroup $ S $ are characterize through int-soft left (resp. right) $ \Gamma $-ideals of $ S. $ Particularly, a $ \Gamma $-regular ordered $ \Gamma $-semigroup $ S $ is a left $ \Gamma $-simple if and only if every int-soft left $ \Gamma $-ideal $ f_{A} $ of $ S $ is a constant function. Also, $ S $ is a semilattice of left (resp. right) $ \Gamma $-simple $ \Gamma $-semigroup if and only if for every int-soft left (resp. right) $ \Gamma $-ideal $ f_{A} $ of $ S, $ $ f_{A}\left(a\right) = f_{A}\left(a\alpha a\right) $ and $ f_{A}\left(a\alpha b\right) = f_{A}\left(b\alpha a\right) $ for all $ a, b\in S $ and $ \alpha \in \Gamma $ hold.

This paper deals with hyperbolic conservation laws exhibiting a flux discontinuity at the origin and which does not admit a weak solution satisfying the Rankine–Hugoniot jump condition. We therefore seek unbounded solutions in the form of shadow waves supported by at the origin. The shadow waves are defined as nets of piecewise constant functions approximating a shock wave to which we add a delta function and possibly another unbounded part.

The paper considers a sequence of solutions to the one-dimensional problem of irreversible deformation of a functionally graded material under conditions of uneven thermal expansion. Numerical solutions are obtained for the problems of heating an elastoplastic sphere, the material constants of which are linear functions of the radius, and exact solutions, in which the material constants are approximated by piecewise constant functions. It is shown that the deformation of a functionally graded elastoplastic material, in which the material constants are specified by piecewise-constant distributions, can be qualitatively described by numerical solutions, in which the material constants are continuous approximations of the corresponding piecewise-constant functions. The obtained numerical and analytical solutions of boundary value problems are graphically analyzed.

We consider the problem of the constancy of the minimizer in the fractional embedding theorem $$\mathcal{H}^s(\Omega) \hookrightarrow L_q(\Omega)$$ for a bounded Lipschitz domain $$\Omega$$ , depending on the domain size. For the family of domains $$\varepsilon \Omega$$ , we prove that, for small dilation coefficients $$\varepsilon$$ , the unique minimizer is constant, whereas for large $$\varepsilon$$ , a constant function is not even a local minimizer. We also discuss whether a constant function is a global minimizer if it is a local one.

We characterize the ANF and the univariate representation of any vectorial function as parts of the ANF and bivariate representation of the Boolean function equal to its graph indicator. We show how this provides, when $F$ is bijective, the expression of $F^{-1}$ and/or allows deriving properties of $F^{-1}$ . We illustrate this with examples and with a tight upper bound on the algebraic degree of $F^{-1}$ by means of that of $F$ . We characterize by the Fourier-Hadamard transform, by the ANF, and by the bivariate representation, that a given Boolean function is the graph indicator of a vectorial function. We also give characterizations of those Boolean functions that are affine equivalent to graph indicators. We express the graph indicators of the sum, product, composition and concatenation of vectorial functions by means of the graph indicators of the functions. We deduce from these results a characterization of the bijectivity of a generic $(n,n)$ -function by the fact that some Boolean function, which appears as a part of the ANF (resp. the bivariate representation) of its graph indicator, is equal to constant function 1. We also address the injectivity of $(n,m)$ -functions. Finally, we study the characterization of the almost perfect nonlinearity of vectorial functions by means of their graph indicators.

Mishra et al. (Applied Mathematics and Computation 237:252–263, 2014) obtained the degree of approximation of $$\tilde{f}$$ , conjugate of a 2π periodic function $$f$$ belonging to the weighted Lipschitz class $$W(L_{r} ,\beta ,\xi )(r \ge 1)$$ , through the Cesaro-Norlund means of the conjugate Fourier series of f, which is an extension of the results of Lal (Applied Mathematics and Computation 209:346–350, 2009) and Singh et al. (International Journal of Mathematics and Mathematical Sciences 2012:1–12, 2012). Zhang (Applied Mathematics and Computation 247:1139–1140, 2014) has pointed out that the conclusions of the theorem of Mishra et al. (Applied Mathematics and Computation 237:252–263, 2014) hold only for constant functions, and thus the results are trivial. In this paper, we redefine the problems of Lal (Applied Mathematics and Computation 209:346–350, 2009) and Mishra et al. (Applied Mathematics and Computation 237:252–263, 2014) for $$f \in W(L_{r} ,\beta ,\xi )(r \ge 1)$$ and its conjugate $$\tilde{f}$$ . We obtain the degree of approximation through more general summability means under weighted norm which in turn resolve the issues raised by Zhang (Applied Mathematics and Computation 247:1139–1140, 2014). We also derive some corollaries from our results.

This paper considers the exponential almost sure (EAS) stability and stabilization problems of continuous-time Markovian jump systems (MJSs) with alternative and continuous controller failures. By introducing a piecewise constant function, the dwell times of a controller useful or not are modeled to be time-variant, whose bounds are also included. Then, sufficient conditions for EAS stability are established by using a method on its stochastic transfer matrix. Moreover, the statistically transient properties of Markov process are considered, and the control gains is gotten by solving some Lyapunov equations. Two examples are used to demonstrate the effectiveness and superiority of the proposed methods.

A heterostructure composed of two parallel homogeneous layers is studied in the limit as their width and the distance between them shrinks to zero simultaneously. The problem is considered in one dimension and the squeezing potential in the Schrodinger equation is chosen in the form of a piecewise constant function. As a result, two families of point interactions with bound state energy are realized from this structure. The specific feature of these interactions is the resonant-tunneling transmission of electrons through one-point singular potentials under certain conditions described by transcendental equations. The solutions to these equations define so-called resonance sets of Lebesgue’s measure zero. A particular example is the potential in the form of the derivative of Dirac’s delta function. For a whole family of point interactions including this example, the existence of a bound state is proven, contrary to the widespread opinion on the non-existence of bound states in δ'-like systems.

In this work, we present a Durrmeyer-type operator having the basis functions in summation and integration due to Aral (Math Commun 24:119–131, 2019) and Pǎltǎnea (Carpath J Math 24(3):378–385, 2008) that preserve the constant functions. We compute the rate of convergence of these operators in a weighted space and also computed a quantitative Voronovskaja-type asymptotic formula as well as a Gruss-Voronovskaya-type approximation.

In this paper we introduce a new conformable derivative call it mixed conformable partial derivative, which obeys classical properties, including linearity, product rule, quotient rule and vanishing derivatives for constant functions. As an application, we establish an Opial type inequality for the mixed second order conformable partial derivatives.

An Archimedean copula is characterised by its generator. This is a real function whose inverse behaves as a survival function. We propose a semiparametric generator based on a quadratic spline. This is achieved by modelling the first derivative of a hazard rate function, in a survival analysis context, as a piecewise constant function. Convexity of our semiparametric generator is obtained by imposing some simple constraints. The induced semiparametric Archimedean copula produces Kendall’s tau association measure that covers the whole range ( − 1 , 1 ) . Inference on the model is done under a Bayesian approach and for some prior specifications we are able to perform an independence test. Properties of the model are illustrated with a simulation study as well as with a real dataset.