Rational Expressions Research Articles

Feedback control for plants with time delay

Abstract The paper considers the problem of feedback control design for plants with time delay. The role of the feedback is presented showing its ability to restrict the uncertainty in the plant model. The conditional feedback control structure is introduced and its properties for restricting model uncertainty are discussed. The presence of local feedback makes the conditional feedback structure pertinent for its usage when the plant contains a time delay element. The time delay model is replaced by its Pade series approximation. The key idea is to consider the time delay model as perturbation around its Pade series approximation. This fact allows utilizing the rational function expression in the procedure of controller design, thus avoiding using the irrational function of the time delay for this purpose. It is shown that the conditional feedback control structure gives an alternative than the Smith predictor method for controller design of plants with time delay e −θs in their structure. The efficiency of the presented method is demonstrated by a numerical example.

Incarnation and decay: reconciling the organizational decision-making and organizational institutional theory perspectives on budgetary research

PurposeThe organizational decision-making perspective (ODM) has a legacy regarding its concern for budgeting as an essential organizational routine in decision-making. Budgeting has also become a direct concern to organizational institutional theory (OIT) because of its prominent role in institution building, where budgeting can build trust in inter-organizational relationships. This paper builds on these two perspectives to explore organizational budget processes' formation, disruption, and re-creation over time.Design/methodology/approachWe conducted a comprehensive review and critical analysis of the ODM and OIT perspectives, focusing on a fundamental paradox between ODM's emphasis on stability through organizational routines and OIT's focus on organizational legitimacy through the decoupled expression of organizational values. We then expanded on these paradoxical concerns in the context of budgeting, formalizing them into specific research propositions for future studies.FindingsTensions around the stability, decay, and re-creation of budgets as organizational routines emerge as a pressing issue requiring further empirical investigation from the ODM perspective. A critical issue in the OIT perspective is the potential for organizational budgets to provide an opportunity to decouple from practice through routinized expressions of rationality and to facilitate loose coupling in practice. These findings offer a fresh perspective and open up new avenues for future research in this area.Originality/valueThis paper contributes to the accounting and organizational research literature by shedding light on how organizations respond to the potential decay of budget routines and the manifestation of organizational values in decoupling processes by further re-creating and elaborating budget processes.

Cubic rational expressions over a finite field

We study and partially classify cubic rational expressions over a finite field [Formula: see text], up to equivalence given by composition with independent Möbius transformations on each side. When [Formula: see text] is even we obtain a full classification. When [Formula: see text] is odd we restrict our classification to expressions with at most three ramification points. However, we prove a general upper bound of [Formula: see text] for the total number of equivalence classes.

Bifurcation analysis and soliton structures of the truncated [formula omitted]-fractional Kuralay-II equation with two analytical techniques

The truncated M-fractional Kuralay (TMFK)-II equation is prevalent in the exploration of specific complex nonlinear wave phenomena. Such types of wave phenomena are more applicable in science and engineering. These equations could potentially provide insights into understanding the intricate dynamics of optical phenomena, encompassing solitons, nonlinear effects, and wave interactions. This study aims to uncover a diverse range of soliton solutions to the model, spanning trigonometric, hyperbolic, exponential, and rational expressions. These solutions are unveiled through the application of extended hyperbolic functions and improved F-expansion techniques, representing the primary objective of this research. The three-dimensional (3D) and two-dimensional (2D) combined charts are plotted for some of these solutions. The impact of the fractional parameters and time variations is also illustrated. Moreover, the models are converted into a planar dynamical system using a Galilean transformation, and the analysis of bifurcation is examined. This research underscores the versatility of the aforementioned techniques for exploring complex nonlinear phenomena across various engineering and scientific disciplines. Finally, the findings of this study hold significant implications for advancing our understanding and analysis of nonlinear wave dynamics in diverse physical systems.

The volume of an isocanted cube is a determinant*

In any dimension d ≥ 2 , we give exact volume formulas of two mutually polar dual convex d-polytopes. The primal body is called isocanted cube of dimension d, depending on two real parameters 0 < a < ℓ . The limit case a = 0 yields a d-cube of edge-length ℓ. We prove that the volume of such a body is the determinant of the matrix of order d having diagonal entries equal to ℓ and a elsewhere. We also compute the volume of the polar dual body, getting a rational expression in ℓ , a , homogeneous of degree − d with rational coefficients. Isocanted cubes are origin-symmetric zonotopes. Zonoids (defined as the limits of families of zonotopes) satisfy the Mahler conjecture; in particular, zonotopes do. Nonetheless, we confirm (by elementary methods) that the Mahler conjecture holds for isocanted cubes.

Arithmetic of D-algebraic functions

We are concerned with the arithmetic of solutions to ordinary or partial nonlinear differential equations which are algebraic in the indeterminates and their derivatives. We call these solutions D-algebraic functions, and their equations are algebraic (ordinary or partial) differential equations (ADEs). The general purpose is to find ADEs whose solutions contain specified rational expressions of solutions to given ADEs. For univariate D-algebraic functions, we show how to derive an ADE of smallest possible order. In the multivariate case, we introduce a general algorithm for these computations and derive conclusions on the order bound of the resulting algebraic PDE. Using our accompanying Maple software, we discuss applications in physics, statistics, and symbolic integration.

FIRE 6.5: Feynman integral reduction with new simplification library

FIRE is a program which performs integration-by-parts (IBP) reductions of Feynman integrals. Originally, the C++ version of FIRE relies on the computer algebra system Fermat by Robert Lewis to simplify rational functions. We present an upgrade of FIRE which incorporates a new library FUEL initially described in a separate publication, which enables a flexible choice of third-party computer algebra systems as simplifiers, as well as efficient communications with some of the simplifiers as C++ libraries rather than through Unix pipes. We achieve significant speedups for IBP reductions of Feynman integrals involving many kinematic variables, when using an open source backend based on FLINT newly added in this work, or the Symbolica backend developed by Ben Ruijl as a potential successor of FORM. Program summaryProgram title: FIRE, version 6.5 (FIRE 6.5)CPC Library link to program files:https://doi.org/10.17632/cy6k69pb3y.2Developer's repository link:https://gitlab.com/feynmanintegrals/fire.gitLicensing provisions: GPLv2Programming language:Wolfram Mathematica 8.0 or higher, C++17Supplementary material: See linked repository for installation instructions.Journal reference of previous version: Comput. Phys. Commun. 247 (2020) 106877Does the new version supersede the previous version?: Yes.Reasons for the new version: The new version no longer relies on a single computer algebra system, Fermat [1], but instead allows a flexible choice of several systems, some of which offer higher performance, especially when the number of variables is large.Summary of revisions: A new library FUEL [2] is used as a core component of the new version of FIRE to access different computer algebra systems as simplifiers of rational function expressions. Since the first release of FUEL described elsewhere, FUEL version 1.0 here has been enhanced with a new backend based on the open source library FLINT [3] that provides highly performant simplification of rational functions.Nature of problem: Feynman integrals of a given family are reduced to a finite set of master integrals, by solving linear equations arising from integration by parts, using Gaussian elimination. The coefficients of the linear equations are generally rational functions in kinematic variables and the spacetime dimension, and the simplification of such rational functions during Gaussian elimination is a key task that is improved in this upgrade of FIRE.Solution method: Computer algebra systems with state-of-the-art capabilities for polynomial GCD computations are used as simplification backends, or simplifiers in short. Due to the design of FIRE, text strings are used as the exchange format for rational functions before and after simplification. A fast C++ parser is written to parse strings into the internal format of an external simplifer, FLINT [3], with state-of-the-art performance for multivariate polynomial calculations. Similarly, the simplifier Symbolica [4] has high performance in both GCD computations and parsing, and has been integrated into FIRE.

Extension of Synthetic Division and Its Applications

This study focused on \enquote{Extension of synthetic division and its applications}. The study was designed to show synthetic division and its extension and to point out the applications of synthetic division and its extension. The study found out that the concepts of polynomial and rational expressions in single variables are basic concepts to deal extension of synthetic division and its applications. Using the preliminary concepts, the concept of synthetic division is extended in this study. Also, the study found out that an extension of synthetic division is used for finding the oblique asymptote of the graph of a rational function, evaluating the integration of some rational functions, representing polynomial expression by factorial function in numerical analysis, and so on.

Abundant New Optical Soliton Solutions to the Biswas–Milovic Equation with Sensitivity Analysis for Optimization

This study extensively explores the Biswas–Milovic equation (BME) with Kerr and power law nonlinearity to extract the unique characteristics of optical soliton solutions. These optical soliton solutions have different applications in the field of precision in optical switching, applications in waveguide design, exploration of nonlinear optical effects, imaging precision, reduced intensity fluctuations, suitability for optical signal processing in optical physics, etc. Through the powerful (G′/G, 1/G)-expansion analytical method, a variety of soliton solutions are expressed in three distinct forms: trigonometric, hyperbolic, and rational expressions. Rigorous validation using Mathematica software ensures precision, while dynamic visual representations vividly portray various soliton patterns such as kink, anti-kink, singular soliton, hyperbolic, dark soliton, and periodic bright soliton solutions. Indeed, a sensitivity analysis was conducted to assess how changes in parameters affect the exact solutions, aiding in the understanding of system behavior and informing decision-making, especially in accurately designing or analyzing real-world optical phenomena. This investigation reveals the significant influence of parameters λ, τ, c, B, and Κ on the precise solutions in Kerr and power law nonlinearities within the BME. Notably, parameter λ exhibits consistently high sensitivity across all scenarios, while parameters τ and c demonstrate pronounced sensitivity in scenario III. The outcomes derived from this method are distinctive and carry significant implications for the dynamics of optical fibers and wave phenomena across various optical systems.

Applications of Series in Calculus: Approximating Radical Functions Through Rational Expressions

Applications of Series in Calculus: Approximating Radical Functions Through Rational Expressions

Research on Delamination Damage Quantification Detection of CFRP Bending Plate Based on Lamb Wave Mode Control.

The carbon-fiber-reinforced polymer (CFRP) bending structure is widely used in aviation. The emergence and spread of delamination damage will decrease the safety of in-service bending structures. Lamb waves can effectively identify delamination damage as a high-damage-sensitivity detection tool. For this present study, the signal difference coefficient (SDC) was introduced to quantify delamination damage and evaluate the sensitivity of A0-mode and S0-mode Lamb waves to delamination damage. The simulation results show that compared with the S0-mode Lamb wave, the A0-mode Lamb wave exhibits higher delamination damage sensitivity. The delamination damage can be quantified based on the strong correlation between the SDC and the delamination damage size. The control effect of the linear array PZT phase time-delay method on the Lamb wave mode was investigated by simulation. The phase time-delay method realizes the generation of a single-mode Lamb wave, which can separately excite the A0-mode and S0-mode Lamb wave to identify delamination damage of different sizes. The A0-mode Lamb wave was excited by the developed one-dimensional miniaturized linear comb transducer (LCT), which was used to conduct the detection experiment on the CFRP bending plate with delamination damage sizes of Φ6.0 mm, Φ10.0 mm, and Φ15.0 mm. The experimental results verify the correctness of the simulation. According to the Hermite interpolation results of the finite-element simulation data, the relationship between the delamination damage size and the SDC was fitted by the Gaussian function and Rational function, which can accurately quantify the delamination damage. The absolute error of the delamination damage quantification with Gaussian and Rational fitting expression does not exceed 0.8 mm and 0.7 mm, and the percentage error is not more than 8% and 7%. The detection and signal processing methods employed in the present research are easy to operate and implement, and accurate delamination damage quantification results have been obtained.

On Suzuki-Proinov Type Contractions in Modular $b$-Metric Spaces with an Application

In this paper, by taking ${{\mathcal C}_\mathcal{A}}-$simulation function and Proinov type function into account, we set up a new contraction mapping called Suzuki$-$Proinov $\mathpzc{Z^*}_{\aE^*}^{\aR}(\alpha)-$contraction, including both rational expressions that possess quadratic terms and $\aE-$type contractions. Furthermore, we demonstrate a common fixed point theorem through the mappings endowed with triangular $\alpha-$admissibility in the setting of modular $b-$metric spaces. Besides that, we achieve some new outcomes that contribute to the current ones in the literature through the main theorem, and, as an application, we examine the existence of solutions to a class of functional equations emerging in dynamic programming.

Stable expression of HIV-1 MPER extended epitope on the surface of the recombinant probiotic bacteria Escherichia Coli Nissle 1917 using CRISPR/Cas9

BackgroundMucosal vaccines have the potential to induce protective immune responses at the sites of infection. Applying CRISPR/Cas9 editing, we aimed to develop a probiotic-based vaccine candidate expressing the HIV-1 envelope membrane-proximal external region (MPER) on the surface of E. coli Nissle 1917.ResultsThe HIV-1 MPER epitope was successfully introduced in the porin OmpF of the E. coli Nissle 1917 (EcN-MPER) and the modification was stable over 30 passages of the recombinant bacteria on the DNA and protein level. Furthermore, the introduced epitope was recognized by a human anti-HIV-1 gp41 (2F5) antibody using both live and heat-killed EcN-MPER, and this antigenicity was also retained over 30 passages. Whole-cell dot blot suggested a stronger binding of anti-HIV-1 gp41 (2F5) to heat-killed EcN-MPER than their live counterpart. An outer membrane vesicle (OMV) – rich extract from EcN-MPER culture supernatant was equally antigenic to anti-HIV-1 gp41 antibody which suggests that the MPER antigen could be harboured in EcN-MPER OMVs. Using quantitative ELISA, we determined the amount of MPER produced by the modified EcN to be 14.3 µg/108 cfu.ConclusionsThe CRISPR/Cas9 technology was an effective method for establishment of recombinant EcN-MPER bacteria that was stable over many passages. The developed EcN-MPER clone was devoid of extraneous plasmids and antibiotic resistance genes which eliminates the risk of plasmid transfer to animal hosts, should this clone be used as a vaccine. Also, the EcN-MPER clone was recognised by anti-HIV-1 gp41 (2F5) both as live and heat-killed bacteria making it suitable for pre-clinical evaluation. Expression of OmpF on bacterial surfaces and released OMVs identifies it as a compelling candidate for recombinant epitope modification, enabling surface epitope presentation on both bacteria and OMVs. By applying the methods described in this study, we present a potential platform for cost-effective and rational vaccine antigen expression and administration, offering promising prospects for further research in the field of vaccine development.

Rational Contraction in Metric Space and Common Fixed Point Theorems

The study of contraction mappings in fixed-point theory is a fascinating and crucial field of mathematics. The concept of contraction plays a vital role in proving the existence and uniqueness of fixed points. Banach's contraction theory offers a fixed point theorem that is widely accepted as unique in most analyses. By using rational expressions in metric spaces, we can achieve unique results in general contraction mapping. These results are based on several innovative ideas stemming from the latest research. The delivered results upgrade and federate many existing outcomes on the topic in the literature Bhardwaj, R. et al. (2007) Chouhan et al. (2014) and Garg and Priyanka (2016). Also gives some suitable examples for verifying our results.

Students’ common errors in quadratic equations: Towards improved mathematics performance

With the aim to close the performance gap between the high performing and the low performing students, this descriptive qualitative study was conducted to analyze students’ common and persisting errors in quadratic equations. Forty-six (N=46) Grade 9 students in a public high school in the Philippines participated in the study. Homework was given where students received instructions from video recordings and other online learning materials using blended learning. The common errors students committed were not following directions, mishandling signs, difficulty in recognizing a quadratic equation, inability to distinguish between solving a quadratic equation and simplifying an algebraic expression, failure to express quadratic equations in standard form, disregarding the negative roots, computational errors in basic algebraic conventions in simplifying radical and rational expressions, factoring, performing special products, and in completing the square. Students were interviewed to validate the error analysis. Immediate feedback through a whole class discussion was conducted the following day to discuss the errors and mistakes committed by students in order to address these and help them learn the necessary concepts and skills in quadratic equations. Instructional strategies and interventions for teaching quadratic equations are recommended for future studies.

Optical fractals and Hump soliton structures in integrable Kuralay-Ⅱ system

&lt;p&gt;The integrable Kuralay-Ⅱ system (K-IIS) plays a significant role in discovering unique complex nonlinear wave phenomena that are particularly useful in optics. This system enhances our understanding of the intricate dynamics involved in wave interactions, solitons, and nonlinear effects in optical phenomena. Using the Riccati modified extended simple equation method (RMESEM), the primary objective of this research project was to analytically find and analyze a wide range of new soliton solutions, particularly fractal soliton solutions, in trigonometric, exponential, rational, hyperbolic, and rational-hyperbolic expressions for K-IIS. Some of these solutions displayed a combination of contour, two-dimensional, and three-dimensional visualizations. This clearly demonstrates that the generated solitons solutions are fractals due to the instability produced by periodic-axial perturbation in complex solutions. In contrast, the genuine solutions, within the framework of K-IIS, take the form of hump solitons. This work demonstrates the adaptability of the K-IIS for studying intricate nonlinear phenomena in a wide range of scientific and practical disciplines. The results of this work will eventually significantly influence our comprehension and analysis of nonlinear wave dynamics in related physical systems.&lt;/p&gt;

COMMON FIXED POINTS OF SUZUKI TYPE Z−CONTRACTION OF TWO PAIRS OF SELFMAPS WITH A RATIONAL EXPRESSION VIA Ψ−SIMULATION FUNCTION

In this paper, we define Suzuki type Z−contraction with a rational expression via Ψ−simulation function and prove the existence and uniqueness of common fixed points of two pairs of selfmaps by using reciprocal continuity and weakly compatible property. An example is provided in support of our result.

Flux optimization using multiple promoters in Halomonas bluephagenesis as a model chassis of the next generation industrial biotechnology

Predictability and robustness are challenges for bioproduction because of the unstable intracellular synthetic activities. With the deeper understanding of the gene expression process, fine-tuning has become a meaningful tool for biosynthesis optimization. This study characterized several gene expression elements and constructed a multiple inducible system that responds to ten different small chemical inducers in halophile bacterium Halomonas bluephagenesis. Genome insertion of regulators was conducted for the purpose of gene cluster stabilization and regulatory plasmid simplification. Additionally, dynamic ranges of the multiple inducible systems were tuned by promoter sequence mutations to achieve diverse scopes for high-resolution gene expression control. The multiple inducible system was successfully employed to precisely control chromoprotein expression, lycopene and poly-3-hydroxybutyrate (PHB) biosynthesis, resulting in colorful bacterial pictures, optimized cell growth, lycopene and PHB accumulation. This study demonstrates a desirable approach for fine-tuning of rational and efficient gene expressions, displaying the significance for metabolic pathway optimization.

Perturbed Gerdjikov–Ivanov equation: Soliton solutions via Backlund transformation

The perturbed Gerdjikov–Ivanov (pGI) equation, a mathematical model that depicts the behaviour of optical pulses during propagation while accounting for perturbation influences, has been thoroughly studied by us. This equation has significant applications in the field of optical fibres, notably for photonic crystal fibres. In this paper, we combined the Riccati-Bernoulli sub-ODE method with the Backlund transformation to construct travelling wave solutions for the perturbed Gerdjikov–Ivanov (pGI) equation. This technique offered us families of solutions as well as a fresh way of dealing with nonlinear models. The results that emerge could greatly advance our knowledge of the physical consequences that are present in the nonlinear model that we are studying. The resulting solutions broaden the scope of earlier findings made using various approaches by including solitons, trigonometric functions, and rational expressions. The Backlund transformation used in this study is distinguished by its simplicity and brevity, and it produces results that are noticeably more thorough than those often obtained by alternative methods. Additionally, we create graphical representations of our results using the computational capabilities of the Maple software. In order to do this, the proper parameter values must be chosen, allowing for a visual study of the results. Through the display of two-dimensional graphical depictions, these effects have been clearly explained, permitting a thorough understanding of their physical implications.

Analytical optical solutions to the nonlinear Zakharov system via logarithmic transformation

In the past few years, there has been a growing interest in investigating the search for solitary wave solutions in the realm of nonlinear partial differential equations. This endeavor represents a captivating and challenging area of research. The primary objective of this study is to investigate the nonlinear Zakharov system through a comprehensive analysis that integrates logarithmic transformations and the symbolic structures of exponential functions. The nonlinear Zakharov system holds remarkable importance as a fundamental cornerstone in the field of plasma physics, derived from the profound intellect of esteemed mathematicians and physicists. By examining a diverse range of solution forms including trigonometric, hyperbolic, and rational expressions, this research delves into the complexities of plasma behavior. Notably, the introduction of arbitrarily selected constants imbues these solutions with a multifaceted and dynamic nature. In support of the findings, the article presents several numerical simulations that align with the derived solutions. The utilized methods stand out for being simple, reliable, and capableof creating fresh solutions for nonlinear partial differential equations in the realm of mathematical physics. Other significance of this research lies in its introduction of innovative methodologies previously unexplored in the study of this particular model, thereby broadening the scientific toolkit. Moreover, the versatility of these techniques offers the potential for seamless adaptation in addressing various other nonlinear partial differential equations.