Mathematics — Open Access Journal

In this paper, we introduce and analyze Stepanov uniformly recurrent functions, Doss uniformly recurrent functions and Doss almost-periodic functions in Lebesgue spaces with variable exponents. We investigate the invariance of these types of generalized almost-periodicity in Lebesgue spaces with variable exponents under the actions of convolution products, providing also some illustrative applications to the abstract semilinear integro-differential inclusions in Banach spaces. Full article

Long-term return expectations or predictions play an important role in planning purposes and guidance of long-term investors. Five-year stock returns are less volatile around their geometric mean than returns of higher frequency, such as one-year returns. One would, therefore, expect models using the latter to better reduce the noise and beat the simple historical mean than models based on the former. However, this paper shows that the general tendency is surprisingly the opposite: long-term forecasts over five years have a similar or even better predictive power when compared to the one-year case. We consider a long list of economic predictors and benchmarks relevant for the long-term investor. Our predictive approach consists of adopting and implementing a fully nonparametric smoother with the covariates and the smoothing parameters chosen by cross-validation. We consistently find that long-term forecasting performs well and recommend drawing more attention to it when designing investment strategies for long-term investors. Furthermore, our preferred predictive model did stand the test of Covid-19 providing a relatively optimistic outlook in March 2020 when uncertainty was all around us with lockdown and facing an unknown new pandemic. Full article

The crossing number $\mathrm{cr}\left(G\right)$ of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main goal of the paper is to state the crossing number of the join product $K_{2,3}+C_n$ for the complete bipartite graph $K_{2,3}$ , where $C_n$ is the cycle on n vertices. In the proofs, the idea of a minimum number of crossings between two distinct configurations in the various forms of arithmetic means will be extended. Finally, adding one more edge to the graph $K_{2,3}$ , we also offer the crossing number of the join product of one other graph with the cycle $C_n$ . Full article

The local controlled generalized H-Bézier model is one of the most useful tools for shape designs and geometric representations in computer-aided geometric design (CAGD), which is owed to its good geometric properties, e.g., symmetry and shape adjustable property. In this paper, some geometric continuity conditions for the generalized cubic H-Bézier model are studied for the purpose of constructing shape-controlled complex curves and surfaces in engineering. Firstly, based on the linear independence of generalized H-Bézier basis functions (GHBF), the conditions of first-order and second-order geometric continuity (namely, G¹ and G² continuity) between two adjacent generalized cubic H-Bézier curves are proposed. Furthermore, following analysis of the terminal properties of GHBF, the conditions of G¹ geometric continuity between two adjacent generalized H-Bézier surfaces are derived and then simplified by choosing appropriate shape parameters. Finally, two operable procedures of smooth continuity for the generalized H-Bézier model are devised. Modeling examples show that the smooth continuity technology of the generalized H-Bézier model can improve the efficiency of computer design for complex curve and surface models. Full article

This paper deals with the numerical solutions and convergence analysis for general singular Lane–Emden type models of fractional order, with appropriate constraint initial conditions. A modified reproducing kernel discretization technique is used for dealing with the fractional Atangana–Baleanu–Caputo operator. In this tendency, novel operational algorithms are built and discussed for covering such singular models in spite of the operator optimality used. Several numerical applications using the well-known fractional Lane–Emden type models are examined, to expound the feasibility and suitability of the approach. From a numerical viewpoint, the obtained results indicate that the method is intelligent and has several features stability for dealing with many fractional models emerging in physics and mathematics, using the new presented derivative. Full article

Technological tables are very important in electrical discharge machining to determine optimal operating conditions for process variables, such as material removal rate or electrode wear. Their determination is of great industrial importance and their experimental determination is very important because they allow the most appropriate operating conditions to be selected beforehand. These technological tables are usually employed for electrical discharge machining of steel, but their number is significantly less in the case of other materials. In this present research study, a methodology based on using a fuzzy inference system to obtain these technological tables is shown with the aim of being able to select the most appropriate manufacturing conditions in advance. In addition, a study of the results obtained using a fuzzy inference system for modeling the behavior of electrical discharge machining parameters is shown. These results are compared to those obtained from response surface methodology. Furthermore, it is demonstrated that the fuzzy system can provide a high degree of precision and, therefore, it can be used to determine the influence of these machining parameters on technological variables, such as roughness, electrode wear, or material removal rate, more efficiently than other techniques. Full article

The present paper investigates digital topological properties of an alignment of fixed point sets which can play an important role in fixed point theory from the viewpoints of computational or digital topology. In digital topology-based fixed point theory, for a digital image $(X,k)$ , let $F\left(X\right)$ be the set of cardinalities of the fixed point sets of all k-continuous self-maps of $(X,k)$ (see Definition 4). In this paper we call it an alignment of fixed point sets of $(X,k)$ . Then we have the following unsolved problem. How many components are there in $F\left(X\right)$ up to 2-connectedness? In particular, let $C_k^{n,l}$ be a simple closed k-curve with l elements in ${\mathbb{Z}}^n$ and $X:=C_k^{n,l_1}\vee C_k^{n,l_2}$ be a digital wedge of $C_k^{n,l_1}$ and $C_k^{n,l_2}$ in ${\mathbb{Z}}^n$ . Then we need to explore both the number of components of $F\left(X\right)$ up to digital 2-connectivity (see Definition 4) and perfectness of $F\left(X\right)$ (see Definition 5). The present paper addresses these issues and, furthermore, solves several problems related to the main issues. Indeed, it turns out that the three models $C_{2n}^{n,4}$ , $C_{3^n-1}^{n,4}$ , and $C_k^{n,6}$ play important roles in studying these topics because the digital fundamental groups of them have strong relationships with alignments of fixed point sets of them. Moreover, we correct some errors stated by Boxer et al. in their recent work and improve them (see Remark 3). This approach can facilitate the studies of pure and applied topologies, digital geometry, mathematical morphology, and image processing and image classification in computer science. The present paper only deals with k-connected spaces in DTC (see ^{Section 2}). Moreover, we will mainly deal with a set X such that $X^{♯}\ge 2$ . Full article

In this paper, we study the regularity of weak solutions to the incompressible Boussinesq equations in ${\mathbb{R}}^3\times (0,T)$ . The main goal is to establish the regularity criterion in terms of one velocity component and the gradient of temperature in Lorentz spaces. Full article

In this work, a kind of normal partner curves of a pseudo null curve on dual space forms is defined and studied. The Frenet frames and curvatures of a pseudo null curve and its associate normal curve on de-Sitter space, its associate normal curve on hyperbolic space, are related by some particular function and the angles between their tangent vector fields, respectively. Meanwhile, the relationships between the normal partner curves of a pseudo null curve are revealed. Last but not least, some examples are given and their graphs are plotted by the aid of a software programme. Full article

The authors establish a set of six new theta-function identities involving multivariable R-functions which are based upon a number of q-product identities and Jacobi’s celebrated triple-product identity. These theta-function identities depict the inter-relationships that exist among theta-function identities and combinatorial partition-theoretic identities. Here, in this paper, we consider and relate the multivariable R-functions to several interesting q-identities such as (for example) a number of q-product identities and Jacobi’s celebrated triple-product identity. Various recent developments on the subject-matter of this article as well as some of its potential application areas are also briefly indicated. Finally, we choose to further emphasize upon some close connections with combinatorial partition-theoretic identities and present a presumably open problem. Full article

This paper discusses the torque data during Parachute Landing Fall (PLF) activity on the sagittal plane by applying Kane’s method. The value of torque is determined in order to identify the movement of extension and flexion at every joint-segment on the parachutist during landing. Data were obtained from three professional and eighteen amateur parachutists, each with three consecutive landings. Quintic Biomechanics Software v26 was selected to capture motion analysis. The mathematical model for the PLF technique was presented based on a two-link kinematics open chain in a two-dimensional space using Kane’s method. The t-test result showed the p-value of torque at each joint between professionals and amateurs (p ≤ 0.05). According to the torque result, the professional parachutists extended their arm then flexion their elbow, shoulder, hip, knee and the ankle plantar flexion during the foot strike phase. The professional demonstrated a perfect PLF technique by identifying the flexion and extension on each joint segment that was involved during landing activity. The value of torque at each joint segment from professional parachutists may be used as a guideline for amateurs to perform optimal landing and minimise the injury. Full article

In this paper, we discuss the averaging method for periodic systems of second order and the behavior of solutions that intersect a hyperplane. We prove an averaging theorem for impact systems. This allows us to investigate the approximate dynamics of mechanical systems, such as the weakly nonlinear and weakly periodically forced Duffing’s equation of a hard spring with an impact wall, or a weakly nonlinear and weakly periodically forced inverted pendulum with double impacts. Full article

Based on the eigenvalue idea and the time-varying weighted vector norm in the state space ${\mathbb{R}}^n$ we construct here the lower and upper bounds of the solutions of uniformly asymptotically stable linear systems. We generalize the known results for the linear time-invariant systems to the linear time-varying ones. Full article

Recently, block backward differentiation formulas (BBDFs) are used successfully for solving stiff differential equations. In this article, a class of hybrid block backward differentiation formulas (HBBDFs) methods that possessed A-stability are constructed by reformulating the BBDFs for the numerical solution of stiff ordinary differential equations (ODEs). The stability and convergence of the new method are investigated. The methods are found to be zero-stable and consistent, hence the method is convergent. Comparisons between the proposed method with exact solutions and existing methods of similar type show that the new extension of the BBDFs improved the stability with acceptable degree of accuracy. Full article

The present study emphasizes the efficacy of a biosurfactant-producing bacterial strain Klebsiella sp. KOD36 in biodegradation of azo dyes and hexavalent chromium individually and in a simultaneous system. The bacterial strain has exhibited a considerable potential for biodegradation of chromium and azo dyes in single and combination systems (maximum 97%, 94% in an individual and combined system, respectively). Simultaneous aerobic biodegradation of azo dyes and hexavalent chromium (SBAHC) was modeled using machine learning programming, which includes gene expression programming, random forest, support vector regression, and support vector regression-fruit fly optimization algorithm. The correlation coefficient includes the dispersion index, and the Willmott agreement index was employed as statistical metrics to assess the performance of each model separately. In addition, the Taylor diagram was used to further investigate the methods used. The findings of the present study were that the support vector regression-fruitfly optimization algorithm (SVR-FOA) with correlation coefficient (CC) of 0.644, (scattered index) SI of 0.374, and (Willmott’s index of agreement) WI of 0.607 performed better than the autonomous support vector regression (SVR), gene expression programming (GEP), and random forest (RF) methods. In addition, the standalone SVR model with CC of 0.146, SI of 0.473, and WI of 0.408 ranked the second best. In summary, the SBAHC can be accurately estimated using the hybrid SVR-FOA method. In other words, FOA has proven to be a powerful optimization algorithm for increasing the accuracy of the SVR method. Full article

The present work highlights the stagnation point flow with mixed convection induced by a Riga plate using a Cu-Al ${}_2$ O ${}_3$ /water hybrid nanofluid. The electromagnetohydrodynamic (EMHD) force generated from the Riga plate was influential in the heat transfer performance and applicable to delay the boundary layer separation. Similarity transformation was used to reduce the complexity of the governing model. MATLAB software, through the bvp4c function, was used to compute the resulting nonlinear ODEs. Pure forced convective flow has a distinctive solution, whereas two similarity solutions were attainable for the buoyancy assisting and opposing flows. The first solution was validated as the physical solution through the analysis of flow stability. The accretion of copper volumetric concentration inflated the heat transfer rate for the aiding and opposing flows. The heat transfer rate increased approximately up to an average of 10.216% when the copper volumetric concentration increased from 0.005 $(0.5\%)$ to 0.03 $(3\%)$ . Full article

The necessity of coordination among entities is essential for the success of any supply chain management (SCM). This paper focuses on coordination between two players and cost-sharing in an SCM that considers a vendor and a buyer. For random demand and complex product production, a flexible production system is recommended. The study aims to minimize the total SCM cost under stochastic conditions. In the flexible production systems, the production rate is introduced as the decision variable and the unit production cost is minimum at the obtained optimal value. The setup cost of flexible systems is higher and to control this, a discrete investment function is utilized. The exact information about the probability distribution of lead time demand is not available with known mean and variance. The issue of unknown distribution of lead time demand is solved by considering a distribution-free approach to find the amount of shortages. The game-theoretic approach is employed to obtain closed-form solutions. First, the model is solved under decentralized SCM based on the Stackelberg model, and then solved under centralized SCM. Bargaining is the central theme of any business nowadays among the players of an SCM to make their profit within a centralized and decentralized setup. For this, a cost allocation model for lead time crashing cost based on the Nash bargaining model with the satisfaction level of SCM members is proposed. The cost allocation model under Nash bargaining achieves exciting results in SCM coordination. Full article

Models of set theory are defined, in which nonconstructible reals first appear on a given level of the projective hierarchy. Our main results are as follows. Suppose that $n\ge 2$ . Then: 1. If it holds in the constructible universe $\mathbf{L}$ that $a\subseteq \omega$ and $a\notin {\Sigma }_n^1\cup {\Pi }_n^1$ , then there is a generic extension of $\mathbf{L}$ in which $a\in {\Delta }_{n+1}^1$ but still $a\notin {\Sigma }_n^1\cup {\Pi }_n^1$ , and moreover, any set $x\subseteq \omega$ , $x\in {\Sigma }_n^1$ , is constructible and ${\Sigma }_n^1$ in $\mathbf{L}$ . 2. There exists a generic extension $\mathbf{L}$ in which it is true that there is a nonconstructible ${\Delta }_{n+1}^1$ set $a\subseteq \omega$ , but all ${\Sigma }_n^1$ sets $x\subseteq \omega$ are constructible and even ${\Sigma }_n^1$ in $\mathbf{L}$ , and in addition, $\mathbf{V}=\mathbf{L}\left[a\right]$ in the extension. 3. There exists an generic extension of $\mathbf{L}$ in which there is a nonconstructible ${\Sigma }_{n+1}^1$ set $a\subseteq \omega$ , but all ${\Delta }_{n+1}^1$ sets $x\subseteq \omega$ are constructible and ${\Delta }_{n+1}^1$ in $\mathbf{L}$ . Thus, nonconstructible reals (here subsets of $\omega$ ) can first appear at a given lightface projective class strictly higher than ${\Sigma }_2^1$ , in an appropriate generic extension of $\mathbf{L}$ . The lower limit ${\Sigma }_2^1$ is motivated by the Shoenfield absoluteness theorem, which implies that all ${\Sigma }_2^1$ sets $a\subseteq \omega$ are constructible. Our methods are based on almost-disjoint forcing. We add a sufficient number of generic reals to $\mathbf{L}$ , which are very similar at a given projective level n but discernible at the next level $n+1$ . Full article

Cavity flow past an obstacle in the presence of an inflow vorticity is considered. The proposed approach to the solution of the problem is based on replacing the continuous vorticity with its discrete form in which the vorticity is concentrated along vortex lines coinciding with the streamlines. The flow between the streamlines is vortex free. The problem is to determine the shape of the streamlines and cavity boundary. The pressure on the cavity boundary is constant and equal to the vapour pressure of the liquid. The pressure is continuous across the streamlines. The theory of complex variables is used to determine the flow in the following subregions coupled via their boundary conditions: a flow in channels with curved walls, a cavity flow in a jet and an infinite flow along a curved wall. The numerical approach is based on the method of successive approximations. The numerical procedure is verified considering a body with a sharp edge, for which the point of cavity detachment is fixed. For smooth bodies, the cavity detachment is determined based on Brillouin’s criterion. It is found that the inflow vorticity delays the cavity detachment and reduces the cavity length. The results obtained are compared with experimental data. Full article