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Abstract: A left canonical factorization theorem for rational matrix functions relative to the unit circle is presented. The result is a time invariant version of a recent strict LU factorization theorem for certain semi-separable operators, due to Dewilde (2012) [6]. Explicit formulas for the factors are also given. The theorem is proved, first by using the state space method of Dewilde (2012) [6], and next by using an operator theory approach. In both cases the main part of the proof concerns rational matrix functions that are unitary on the unit circle. [Copyright &y& Elsevier]

The present comment concerns the correct presentation of an algorithm proposed in the above paper for the glow-curve deconvolution in the case of continuous distribution of trapping states. Since most researchers would use directly the proposed algorithm as published, they should be notified of its correct formulation during the fitting of TL glow curves of materials with continuous trap distribution using this Equation. [ABSTRACT FROM AUTHOR]

Abstract: The study used the simulated e-paper to investigate how the bending radius of curvature (−10cm, plane, and 10cm) and 12 text/background color-combinations of e-paper affect subjects’ visual performance and subjective preference under various ambient illuminance conditions (200 and 500lx). Analysis results indicated that the bending curvature and ambient illuminance did not significantly affect subjects’ visual performance. However, subjects visual performance differed significantly for different text/background color combinations of the simulated e-paper. When the background color of the simulated e-paper was set to yellow-like condition and the luminance of the text was low (2.2 and 4.6cd/m2), subjects’ visual performance was best. Regarding the subjective preferences of subjects, the results of this research also demonstrated that the bending curvature, text/background color combinations and ambient illuminance all significantly affected the subjective preferences of subjects. Subjects exhibited the best preference under the following settings: bending curvature of the simulated e-paper set to plane; background color of the simulated e-paper set to yellow-like condition and low text luminance (2.2 or 4.6cd/m2); high ambient illuminance (500lx). [Copyright &y& Elsevier]

• Mathematical and physical analyses of the introduced problem. • Reporting a serious error in an important dimensionless parameter. • Indicating that the reported numerical results are completely erroneous. This is a comment on the paper [Optics and Laser Technology 128 (2020) 106233; https://doi.org/10.1016/j.optlastec.2020.106233 ] titled "Exact coupled solution for photothermal semiconducting beams using a refined multi-phase-lag theory". Herein, we report a serious error in an important dimensionless parameter employed in the problem, revealing that the defined parameter is dimensional and wrong. Accordingly, all given numerical results are based on this serious mistake, and are completely erroneous. [ABSTRACT FROM AUTHOR]

We would like to present the ideas of Makoto Matsumoto and András Rapcsák in the projective theory of Finsler spaces. Finally, we give some remarks using of Matsumoto–Rapcsák Theorem. [ABSTRACT FROM AUTHOR]

Abstract: In the present paper we studied the Voronovskaya type theorem for modified Gamma operators presented by A. İzgi in the paper “Voronovskaya type asymptotic approximation by modified Gamma operators” [Applied Mathematics and Computation 217 (2011) 8061–8067]. We show that this theorem is not true. We present a correct form of this theorem and its proof. Furthermore, we obtain an error estimate for modified Gamma operators. [Copyright &y& Elsevier]

Abstract: Comment on the paper “A class of methods based on non-polynomial spline functions for the solution of a special fourth-order boundary-value problems with engineering applications” in Applied Mathematics and Computation 174 (2006) 1169–1180. The paper considered a class of two-point boundary-value problems of the form where f(x) and g(x) are continuous on [a, b], and A i and B i (i =1,2) are finite real constants. Here we correct some mistake in derivation of non-polynomial spline, boundary formulas, truncation errors, convergence analysis and computational experiments. [Copyright &y& Elsevier]

Abstract: In this paper, we determine the modularities of asymptotically optimal towers found by Maharaj and Wulftange in 2005. This is also a continuation of the work of Elkies in 2002. [Copyright &y& Elsevier]

Abstract: In this paper, we establish new Lyapunov-type inequalities for two classes of one-dimensional quasilinear elliptic systems of resonant type, which improve the recent results of Tang and He [X.H. Tang, X. He, Lower bounds for generalized eigenvalues of the quasilinear systems, J. Math. Anal. Appl. 385 (2012) 72–85] when 1< p i <2 for i =1,2,…, n. [Copyright &y& Elsevier]

Abstract: In this paper, some sufficient conditions are established for the oscillation of second-order neutral differential equations where and . On the other hand, some new oscillation criteria are established for the second-order nonlinear neutral delay differential equations where . The results obtained here complement and correct some known results in . Some examples are given to illustrate the main results. [Copyright &y& Elsevier]

Abstract: In this short note an addition to the paper [A.G. Bratsos, M. Ehrhardt, I.Th. Famelis, A discrete Adomian decomposition method for discrete nonlinear Schrödinger equations, Appl. Math. Comput. 197(1) (2008) 190–205] using the modulus of the terms evaluated from the Adomian decomposition method on p. 194 and their relation to the convergence of the resulting series is presented. Conclusions for the accuracy of the approximated solution are derived. [Copyright &y& Elsevier]

Abstract: A generalization of two sharp inequalities in a recent paper by N. Ujević is established. Applications in numerical integration are also given and the results of N. Ujević are revised and improved. [Copyright &y& Elsevier]

Abstract: In this note, we point out an error in the recently published article [R.K. Mohanty, M.K. Jain, D. Dhall, A cubic spline approximation and application of TAGE iterative method for the solution of two-point boundary value problems with forcing function in integral form, Appl. Math. Model. 35 (2011) 3036–3047] and then correct it. [Copyright &y& Elsevier]

Abstract: The aim of this note is to show that the optimization algorithm proposed in may not lead to the optimal solution in some cases. In fact, the optimization problem remains open if we do not apply the branch-and-bound method to solve it or compute all the objective values of minimal solutions of the feasible domain. [Copyright &y& Elsevier]

Abstract: A counterexample is provided in this short note to show that some of local convergence theorems established in [D.K.R. Babajee, M.Z. Dauhoo, An analysis of the properties of the variants of Newton’s method with third order convergence, Appl. Math. Comput. 183 (2006) 659–684] are not always true. Some mistakes in the proofs of these theorems are pointed out. [Copyright &y& Elsevier]

Abstract: In this technical note, we give a short extension application for nonlinear sum of ratios problem (P) considered in [Y.J. Wang, K.C. Zhang, Global optimization of nonlinear sum of ratios problem, Applied Mathematics and Computation 158 (2004) 319–330]. Actually our result is slightly more general, since we do not specify additional positive coefficient for each ratio. In this note, we use different equivalent problem as done in Wang and Zhang (2004). Our method introduce p variables less than other method (Wang and Zhang, 2004), and our approach need not additional special program to obtain the upper and lower bound of numerator and denominator for each ratio in the objective function. [Copyright &y& Elsevier]

Abstract: We study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by L. Carlitz. We show that they are a special case of the Sheffer polynomials and point out some interesting connections with certain Sobolev orthogonal polynomials. [Copyright &y& Elsevier]

Abstract: In this short note we point to an error in the proof of a theorem stated in [L. Balková, E. Pelantová, Š. Starosta, On Brlek-Reutenauer conjecture, Theoret. Comput. Sci. 412 (2011) 5649–5655]. By constructing a counterexample, we show that the assertion of the theorem is actually incorrect. Although this theorem is of a technical character, it was used in an argument leading to a corollary of a general interest to the Brlek-Reutenauer conjecture, and thus as a consequence of this note we have that the proof of the mentioned corollary is also flawed. [Copyright &y& Elsevier]

Abstract: In this note, we will modify several gaps in Gurudwan and Sharma [N. Gurudwan, B.K. Sharma, Convergence theorem for the common solution for a finite family of -strongly accretive operator equations, Appl. Math. Comput. 217 (2011) 6748–6754]. [Copyright &y& Elsevier]

This paper derives well-posedness and asymptotic results that provide qualitative information about the behavior, mechanism and strategies used by living organisms to navigate their biological networks. Chemical driven swimming is a captivating phenomenon that is observed in various living organisms like bacteria and protozoa but the problem in weighted networks is more complex, since the equations of parabolic-parabolic Keller-Segel model coupled with incompressible Navier-Stokes equations must be reformulated in a discrete setting. The starting point is to transpose the coupled system from the Euclidean case to the connected networks with certain network-theoretic simplified approaches, which yields fruitful key results. These results not only enable the construction of global solutions but also serve as a foundation for determining information about the stability and large time behavior of the system. Then, decay rates are well established to predict important features, such as how quickly weak solutions at a given point decrease over time due to dissipative processes. Additionally, the L 1 - convergence of cell densities towards the self-similar Gaussian solution of the heat equation is well proved by time dependent scaling, which shows that the solution maintains its shape and only scales in time and space as time evolves. Finally, this paper includes many numerical tests through a recent robust numerical scheme to illustrate the theoretical results and to develop computational control and prediction of living organisms' trajectories around central nodes in networked flows. [ABSTRACT FROM AUTHOR]

Abstract: In abstract of the paper [A. Rafiq, A note on “A family of methods for solving nonlinear equations”, Appl. Math. Comput. 195 (2008) 819–821] we can find the following sentences. We cite: Ujević et al. introduced a family of methods for solving nonlinear equations. However the main Algorithm 1 put forward by Ujević et al. (p. 7) is wrong. This is the main aim of this note. We also point out some major bugs in the results of Ujević et al. – the end of the citation. Here it is shown that all of the mentioned assertions are not true. In other words, the Algorithm 1 is correct (up to an obvious misprint, which is not mentioned in the above paper) and there are no major bugs in the paper by Ujević et al. In fact, these observations, which will be given in this note, show that the main aim of the paper by Rafiq is wrong. [Copyright &y& Elsevier]

Abstract: It is shown that there are some mistakes in the paper “A singular-value decomposition-based image watermarking using genetic algorithm” by Veysel Aslantas in Int. J. Electron. Commun. AEU 2008;62(5):386–94. [Copyright &y& Elsevier]

Abstract: This special issue contains the revised and extended versions of three papers presented in the 4th Workshop on Advances in Model-Based Testing (A-MOST 2008). In addition to an executive summary of these three papers, this preface briefly reviews the papers published in the four editions of the A-MOST workshop. We hope that the reader will find this special issue interesting and informative. [Copyright &y& Elsevier]

Abstract: A lemma presented by Matkowski requires a proof. [Copyright &y& Elsevier]

Abstract: We recall some bibliography on fast polynomial multiplication related to the recent progress in the paper by Cenk and Ozbudak of 2011. [Copyright &y& Elsevier]

• A chamber model considering pressure distributions of compressors is proposed. • An exhaustive mathematical model and an efficient solution method are provided. • More geometry features and compression mechanisms can be investigated. • Parametric studies are performed using the new model with good trends. Further efficiency improvements of screw compressors rely on computer modeling and mathematical analysis at the design stage. As speed and capacity increase, the uniformity assumption used in the conventional chamber model for twin-screw compressors is no longer suitable. At the same time, CFD simulations are complicated and time-consuming for design optimization. Therefore, in this paper, a chamber model considering pressure distribution is proposed, which is fast enough and more applicable. Each working unit of a screw compressor is considered a one-dimensional compressible transient flow with two moving pistons, which allow the passage of fluid. Numerical methods of the model are verified by a shock wave test. A twin-screw compressor was also simulated and the results of the chamber model were compared with the CFD results. Pressure fluctuations like those in previous experimental studies can be captured in the simulated indicator diagram. General trends with different speeds, suction ports, and wrap angles could also be recognized in the simulation results. The compression process and the influence of structural parameters on the compressor's functioning are revealed intuitively. By combining existing chamber models, the applicability of the new model can be further extended, and it is expected to be a helpful tool for compressor design. [ABSTRACT FROM AUTHOR]

The integration of multiterminal hybrid HVDC grids connecting LCC- and VSC-based networks faces several technical challenges such as DC fault isolation, ensuring multi-vendor interoperability, managing high DC voltage levels, and facilitating high-speed power reversal without interruptions. The two-stage DC-DC converter emerges as a key solution to address these challenges. By implementing the modular multilevel converter (MMC) structure, the converter's basic topology includes half-bridge sub-modules on the VSC side and full-bridge sub-modules on the LCC side. However, while this topology has been discussed in the literature, its connection to an LCC-based network with controlled current magnitude lacks detailed analysis regarding operational challenges, control strategies under various scenarios, and design considerations. This paper fills this gap by providing comprehensive mathematical analysis, design insights, and control strategies for the modular DC-DC converter to regulate DC voltage on the LCC-HVDC side. Additionally, the proposed control scheme minimizes the interfacing inductor between the two bridges, ensuring uninterrupted power flow during reversal and effective handling of DC faults. Validation through Control-Hardware-in-the-Loop testing across diverse operational and fault scenarios, along with a comparative analysis of different converters, further strengthens the findings. © 2017 Elsevier Inc. All rights reserved. [ABSTRACT FROM AUTHOR]

This paper enriches a pre-existing decision algorithm, which in turn augmented a fragment of Tarski's elementary algebra with one-argument real functions endowed with a continuous first derivative. In its present (still quantifier-free) version, our decidable language embodies the addition of functions and multiplication of functions by scalars; the issue we address is the one of satisfiability. As regards real numbers, individual variables and constructs designating the basic arithmetic operations are available, along with comparison relators. As regards functions, we have variables of another sort, out of which compound terms are formed by means of constructs designating addition and differentiation. An array of predicates designates various relationships between functions, as well as function properties, that may hold over intervals of the real line; those are: function comparisons, strict and non-strict monotonicity / convexity / concavity, comparisons between the derivative of a function and a real-valued term. Our decision method consists in preprocessing the given formula into an equi-satisfiable quantifier-free formula of the elementary algebra of real numbers, whose satisfiability can then be checked by means of Tarski's decision method. No direct reference to functions will appear in the target formula, each function variable having been superseded by a collection of stub real variables; hence, in order to prove that the proposed translation is satisfiability-preserving, we must figure out a flexible-enough family of interpolating C 1 functions that can accommodate a model for the source formula whenever the target formula turns out to be satisfiable. With respect to the results announced in earlier papers of the same stream, a significant effort went into designing the family of interpolating functions so that it could meet the new constraints stemming from the presence of function addition (along with differentiation) among the constructs of our fragment of mathematical analysis. • A formal language RDF* concerned with differentiable real functions is proposed. • The class of differentiable functions treated is closed under addition. • The expressive power of RDF* is illustrated through a gallery of examples. • A satisfiability-preserving algorithm reducing RDF* formulas into an existential sentence of Tarskian algebra is presented. • The correctness of the reduction algorithm is reported. [ABSTRACT FROM AUTHOR]

The lack of ultrasensitive biosensors is the major challenge in the control of pandemics at the very initial stage, so it is necessary to develop new biosensors for the prompt treatment, and the piezoelectric biosensor based on ultrasensitive MEMS system offers a good opportunity. This paper presents a novel approach using polyvinylidene fluoride (PVDF) unsmooth nanofibers to transfer electronic current. The unique unsmooth surface of these nanofibers provides a high surface energy (geometrical potential), making them highly sensitive to microorganisms (e.g., viruses) absorbed on their surface. The absorption of viruses has an obvious effect on the nanofiber's displacement and electrical resistivity. This paper presents a mathematical model that demonstrates the system's ultra-sensitivity. This rigorous mathematical analysis shows that even when the pull-in voltage in the PVDF nanofiber changes by just 0.0589 %, a dramatic shift from normal periodic motion to pull-in motion is predicted. This paper provides a new and reliable theoretical way to detect pandemics at the very beginning stage. [Display omitted] • A Polyvinylidene Fluoride(PVDF) nanofiber is used in a micro-electromechanical system. • PVDF nanofiber has good piezoelectric property. • A mathematical model is established to show the extremely sensitiveness of the MEMS-based biosensor. [ABSTRACT FROM AUTHOR]

In this series of papers we aim to provide a mathematically comprehensive framework to the hamiltonian pictures of quantum field theory in curved spacetimes. Our final goal is to study the kinematics and the dynamics of the theory from the point of differential geometry in infinite dimensions. In this first part we introduce the tools of Gaussian analysis in infinite dimensional spaces of distributions. These spaces will serve the basis to understand the Schrödinger and Holomorphic pictures, over arbitrary Cauchy hypersurfaces, using tools of Hida-Malliavin calculus. Here the Wiener-Ito decomposition theorem provides the QFT particle interpretation. Special emphasis is done in the applications to quantization of these tools in the second part of this paper. We devote a section to introduce Hida test functions as a notion of second quantized test functions. We also analyze of the ingredients of classical field theory modeled as distributions paving the way for quantization procedures that will be analyzed in [3]. [ABSTRACT FROM AUTHOR]

The rapid growth in HVDC grids is becoming inevitable for long-distance power transmission. Therefore, the idea of interconnection between the point-to-point links becomes essential. However, these point-to-point connections face several challenges such as the requirement of DC fault blocking capability, interfacing of different grounding schemes, offering multi-vendor interoperability, and difficulty to achieve high DC voltage stepping. DC-DC converters are considered the optimum solution to tackle these challenges in DC grids interconnection. In this paper, a new hybrid modular DC-DC converter is proposed that achieves a low number of semiconductors, low losses, and cost in comparison to other DC-DC converters due to the utilization of thyristors. The new DC-DC converter consists of two hybrid MMC bridges connected through an isolating transformer. Each MMC bridge is comprised of half bridge submodules and bidirectional thyristors. Detailed mathematical analysis, design, and control are illustrated. A comparison is carried out between different topologies in terms of semiconductor count, power loss, and cost. Also, both simulation model and experimental test rig are built to validate the proposed hybrid modular DC-DC converter under different scenarios. Finally, another variant of the hybrid-thyristor based converter (version two) is proposed for multiport DC-Hub application to achieve DC fault blocking without turning off all connected bridges. [ABSTRACT FROM AUTHOR]

This paper focuses on resolving the distributed formation containment control issue in general multi-agent systems (MASs) possessing multiple leader agents subject to limited communication resources. The unnecessary data transmissions among agents in MASs are reduced by utilizing a unique dynamic event-triggering communication mechanism (DETCM), which can build communication time sequences for each agent automatically. Based on the sampled states of agents at event-triggering instants, a novel distributed formation containment control protocol is proposed. By mathematical analysis, the formation containment control problem is turned into the problem of stability in a reduced-order system. The global formation containment of the MASs is finally achieved through the application of Lyapunov-Krasovskii functionals and general inequalities. Moreover, a valid example is also given to support the validity and effectiveness of the findings. • This paper considers MASs with multiple leader agents, which has a broad application range and significant value. • A distributed formation containment control is proposed, which benefits the containment of leader and promotes formation of followers. • The DETM proposed in this paper eliminates the need for real-time communication and reduces the communication load. • A dynamic parameter is introduced to the ETM, which makes the triggering conditions change according to the states of agents. [ABSTRACT FROM AUTHOR]

The paper proposes an ameliorated methodology to minimize the effect of charge injection over a wide input common-mode range. Instead of a conventional single dummy switch compensation [1] , the multi-dummy switches are proposed and employed to eradicate the injected charge on to the sampling capacitor. A detailed methodology is presented to compensate the charge injection in a MOS switch. The closed-form equations are derived mathematically to substantiate the proposed technique. For the proof-of-concept, a track-and-hold (T/H) stage has been simulated in 0. 18 μ m CMOS technology with the proposed technique for a 10-bit resolution. The proposed technique based T/H stage exhibits the spurious free dynamic range (SFDR) of 62.6 dB, effective number of bits (ENOB) of 9.36, peak input-referred third-order intercept point (IIP3) of 13.02 dBm and an input-referred 1 dB compression point (P 1dB) of 3.8 dBm at 1.074 MHz input frequency, sampled at 100 MSa/s. The performance of the proposed method is compared with the existing single dummy switch compensation method where our technique shows 88.8% compensation in minimizing the charge injection error and 272% improvement in dynamic linearity. Moreover, the presented technique quantifies 9 × improvement in mean percentage error (MPE) when simulated across the various process corners and rail-to-rail input common-mode voltage. • The paper presents a multi-dummy switch technique to minimize the charge injection error in a MOS switch. • Mathematical analysis has been carried out to validate the concept. • A design methodology is proposed to limit the charge injection error within ±0.5 LSB for 10-bit track-and-hold circuit. • The presented method is compared with existing single dummy switch compensation technique and the results depict 272% improvement in dynamic linearity. • The proposed method provides 9X improvement in the mean percentage error over the existing method. • The design is simulated in CMOS 180 nm technology and the post-layout simulation results are reported in the paper. [ABSTRACT FROM AUTHOR]

Abstract: This work presents critics'' arguments about the recently defined physical parameter Amlouk-Boubaker Optothermal Expansivity. The detailed definition proposed by Zhang et al. in Materials Letters (No. 64-2009-pp. 778–780) has been based on the minimization of a pondered quadratic function. In the actual study, the uniqueness of the proposed solution is contested on the bases of mathematical evidence, linked to polynomials expansion properties. The authors of the discussed study used a non-weighted evaluation function which involved indifferently two relevant parameters. The suggested corrections along with basic criticism were based on the foundation of optimality rather than those of the optimizing stoichiometry-related protocol itself. [Copyright &y& Elsevier]

Abstract: Two-dimensional axially moving materials have a wide range of industrial applications such as papers, plastics and composites in producing lines, power transmission and conveyor belts, etc. In many of these instances, the moving material is not isotropic, but is a single-layer orthotropic material or consists of several orthotropic layers. In this article, free vibration of axially moving symmetrically laminated plates subjected to in-plan forces is studied by classical plate theory. This category includes symmetric cross-ply and angle-ply laminates and anisotropic plates. Firstly, an exact method is developed to analyze vibration of multi-span traveling cross-ply laminates, and then a semi-analytical finite strip method is extended for moving symmetric laminated plates in general, with arbitrary boundary conditions. By the finite strip method intermediate elastic or rigid supports can also be added to the model of the moving plate. The supports may be in the form of point, line or local distributed supports. [Copyright &y& Elsevier]

Abstract: In the present paper, a description and mathematical analysis of a simple model of nonlinear pattern formation is given. The model is based on the so-called activator-inhibitor system proposed by Thomas. We introduce time delay into the reaction term and focus on its influence on morphogenesis and pattern formation. Numerical simulations are presented and compared for both cases without and with delay. [Copyright &y& Elsevier]

Investigating the effects of periodic magnetic field and triple diffusion on bioconvection flow through a rotating cone populated by oxytactic bacteria and an Eyring-Powell fluid is innovative and significant. The magneto-bioconvection flow of an Eyring-Powell multi-diffusive fluid across a spinning cone is investigated in this paper, considering the effect of oxytactic microorganisms. The non-similar technique is used in the mathematical analysis of the flow over a spinning cone. The flow under consideration contains two diffusive species: liquid hydrogen and liquid oxygen. In light of the periodic magnetic field, the surface gradients, notably skin friction, exhibit wavy effects in the boundary layer domain. The governing equations for the fluid flow in the current flow problem, accompanied by heat diffusion, species diffusion, rotation, bioconvection, and periodic magnetic, are highly coupled nonlinear PDEs dependent on the proper initial and boundary conditions. Mangler's transformations convert them into non-dimensional forms, and numerical non-similar solutions are produced using implicit finite difference approximation and quasi-linearisation. The enriching values of bioconvective Rayleigh number Rb decline the velocity F, as a result, lessen the friction between the surrounding fluid and the cone's surface. The improving Peclet number Pe and microbial density difference σ reduced the microorganism density profile and heightened the microorganism density number R e - 1 / 2 N n. [ABSTRACT FROM AUTHOR]

In this manuscript, we are going to study a novel model of the dynamics of alcohol consumption under induced complications. The mentioned model is considered under the concept of conformable fractional order derivative (CFOD). Currently, most of real-world problems are considered under fractional order derivatives because of their stable and global behavior. First, we will investigate the model for qualitative theory including existence and uniqueness of solution and Ulam-Hyers stability. For qualitative theory, we will use fixed point theory. In addition, we use a numerical method to find the approximate solution of the proposed model. In the final part of the paper, we give a detailed discussion of its numerical results and its graphical presentation. [ABSTRACT FROM AUTHOR]

[Display omitted] • This article proposes a new robust nonlinear controller that stabilizes a chaotic finance system in a finite-time without cancellation of the spacecraft's nonlinear terms, it improves the efficiency of the closed-loop. • It accomplishes an oscillation-free faster convergence of the perturbed state variables to the desired steady-state. • The proposed controller is insensitive to the parameter uncertainties of the nonlinear terms and exogenous disturbances. • The paper performs a comparative study to verify the performance and efficiency of the proposed controller. Robust, stable financial systems significantly improve the growth of an economic system. The stabilization of financial systems poses the following challenges. The state variables' trajectories (i) lie outside the basin of attraction, (ii) have high oscillations, and (iii) converge to the equilibrium state slowly. This paper aims to design a controller that develops a robust, stable financial closed-loop system to address the challenges above by (i) attracting all state variables to the origin, (ii) reducing the oscillations, and (iii) increasing the gradient of the convergence. This paper proposes a detailed mathematical analysis of the steady-state stability, dissipative characteristics, the Lyapunov exponents, bifurcation phenomena, and Poincare maps of chaotic financial dynamic systems. The proposed controller does not cancel the nonlinear terms appearing in the closed-loop. This structure is robust to the smoothly varying system parameters and improves closed-loop efficiency. Further, the controller eradicates the effects of inevitable exogenous disturbances and accomplishes a faster, oscillation-free convergence of the perturbed state variables to the desired steady-state within a finite time. The Lyapunov stability analysis proves the closed-loop global stability. The paper also discusses finite-time stability analysis and describes the controller parameters' effects on the convergence rates. Computer-based simulations endorse the theoretical findings, and the comparative study highlights the benefits. Theoretical analysis proofs and computer simulation results verify that the proposed controller compels the state trajectories, including trajectories outside the basin of attraction, to the origin within finite time without oscillations while being faster than the other controllers discussed in the comparative study section. This article proposes a novel robust, nonlinear finite-time controller for the robust stabilization of the chaotic finance model. It provides an in-depth analysis based on the Lyapunov stability theory and computer simulation results to verify the robust convergence of the state variables to the origin. [ABSTRACT FROM AUTHOR]

• Detailed modeling of CIGRE-14 bus standard test system along with inverter control scheme (with and without grid code) and protection system is done in PSCAD environment. • The performance of directional relaying schemes (32Q and 67pos,67neg,67zero) are evaluated in the presence of inverter-interfaced renewable energy resources. A comparative study on the performance of the relays is done in the presence of inverter-interfaced renewable energy resources for different fault types and fault resistances. • The interdependencies between the inverter control scheme used and performance of relaying schemes have been analyzed through mathematical analysis and circuit realization of the inverter-interfaced renewable energy resources. • Based on the mathematical analysis and circuit realization, the probable causes of relay failures in the presence of inverter-interfaced renewable energy resources are detailed. • The impact analysis of the dynamic response of the controller on the protective relay settings from the perspective of protection operation time-frame has been attempted. The potential impacts of the inverter-interfaced renewable energy resources (IIRERs) on the performance of sequence-based directional relaying schemes (SBDRS) have been investigated in this paper. The directional overcurrent relays such as 67 POS , 67 NEG , 67 ZERO and 32Q have been considered for the investigation. CIGRE-14 bus system with IIRERs along with detailed modeling of inverter control scheme (with and without grid codes) and SBDRS have been done in the PSCAD environment. The mathematical analysis and circuit realization of the IIRERs have been attempted in the paper. The exclusive fault characteristics exhibited by IIRERs cause the conventional SBDRS to mal-operate. Significant change in levels of sequence currents and angular relationships between sequence voltages and currents during faults contributes to relay's unpredictable behavior. The comparative study based on different fault types, fault resistances and interconnecting transformer connections have also been done. The study reveals the impact of non-generalized angular differences between positive sequence voltage and current at the inverter terminal on performance of 67 POS. It is found that 67 NEG and 32Q mal-operated for all types of unbalanced faults while the 67 ZERO can be utilized for grounded fault with delta-star grounded connection of interconnecting transformer. [ABSTRACT FROM AUTHOR]