Algebraic Manipulation Research Articles

A general approach for the derivation of optimal solutions without derivatives

In recent years, there have been a number of approaches in the literature that determine closed form optimal solutions to some classical inventory problems and their various extensions without using derivatives. Earliest of such approaches are based on algebraic manipulations with some a priori assumptions about the optimal solutions or based on properties of the quadratic function, whereas more recent approaches are based on problem specific assumptions such as an infinite planning horizon and integer decision variables, and analysing limiting behaviour of the decision variables and the objective function. None of the approaches is extendable to other similar optimisation problems. In this paper, we propose a general approach without any a priori assumptions of the optimal solution and without any problem specific assumptions. We demonstrate the proposed approach by applying it to some classical problems such as the EOQ and EPQ, with and without planned backorders.

Propuesta para el tratamiento de problemas de tasas de variación relacionadas mediante el uso de GeoGebra: Un estudio de casos

The main objective of this research study is to evaluate an educational strategy specifically designed as an action plan for the topic of related variation rates in the differential calculus on-learning unit during the COVID-19 pandemic. The methodology is based on a qualitative paradigm with constructivist epistemology and on a semiotic focus on teaching and learning mathematics. Videoconferences, teaching sequences, and problems are used. The results show a significant improvement of students in their abilities related to mathematical language and conceptual applications of differential calculus, mainly in the concepts of variation rates and algebraic manipulation. Improvements are also observed in the transfer of semiotic records, independent of the concept of their representations. It is concluded that teaching and learning by using didactic strategies supported with GeoGebra, allow students to conduct research and improve their abstract capacity, their lateral thinking, and their persistence by integrating thinking abilities and conceptual comprehension. © 2021,Formacion Universitaria.All Rights Reserved

Why computational teaching? the advantage of using the maple software in teaching exact sciences

The computational thinking becomes more and more an essential part of modern teaching. In this context, the resources, including the collection of online lessons plan, videos and other resources were created to provide a better understanding of CT. But the most important are software resources, because of their flexibility and wide application tools. The aim of the present paper is to have an introduction trip focusing on the advantages of using powerful software like MAPLE in computational teaching. Some examples are focused on representing the objects and basic algebraic manipulation commands, very successfully used in teaching sciences.

Approximating the Community Structure of the Long Tail

In many social media applications, a small fraction of the members are highly linked while most are sparsely connected to the network. Such a skewed distribution is sometimes referred to as the "long tail". Popular applications like meme trackers and content aggregators mine for information from only the popular blogs located at the head of this curve. On the other hand, the long tail contains large volumes of interesting information and niches. The question we address in this work is how best to approximate the community membership of entities in the long tail using only a small percentage of the entire graph structure. Our technique utilizes basic linear algebra manipulations and spectral methods. It has the advantage of quickly and efficiently finding a reasonable approximation of the community structure of the overall network. Such a method has significant applications in blog analysis engines as well as social media monitoring tools in general.

Attosecond soliton switching through the interactions of two and three solitons in an inhomogeneous fiber

We obtain the exact two and three soliton solutions for higher order NLS equation with variable coefficients using Darboux transformation method with some algebraic manipulations based on constructed Lax Pair. For the first time, switching characteristics of two and three solitons in the attosecond regime via inelastic soliton interactions are discussed through some graphical illustrations. Additionally, effects of the inhomogeneous coefficients on propagation features of solitons are analyzed graphically. Our results have certain applications in the construction of optical switching and soliton management in optical communication systems.

Input–output‐driven gain‐adaptive iterative learning control for linear discrete‐time‐invariant systems

AbstractFor a class of linear discrete‐time‐invariant systems repetitively operated in a finite time length, an iterative correction algorithm is exploited to identify the system Markov parameters with multi‐operation inputs and outputs in obeying a criterion, of which the time order of the corrector is adaptive to the time order of the controller. Interactively, an adaptive iterative learning control is architected which composes the compensator with the approximated Markov parameters and the tracking error in sequentially minimizing a quadratic objective function of the tracking error and the compensation. Algebraic manipulation achieves that the Markov parameters correction error is monotonically declining without any step parameter selection. Further, a rigorous derivation clarifies that the time order of the learning compensator must agree to the relative degree of the system and the tracking error is linearly monotonously convergent if the Markov parameters correction error falls into an allowable range. Inspiringly, the methods and results release the confinement to both the correction and the learning ratios. In addition, a smaller learning ratio factor tolerates a larger Markov parameters correction error. Numerical experiments support the significance and the validity.

Efficient multi-frequency solutions of FE–BE coupled structural–acoustic problems using Arnoldi-based dimension reduction approach

The frequency sweep analysis is indispensable for the performance prediction and optimization design of structural–acoustic interaction problems. The coupled finite element and boundary element (FE–BE) method has been widely used to perform such simulations. However, the straightforward solution of the resulting hybrid model for a large number of frequencies is computationally prohibitive due to the unfavorable properties of involved matrices, e.g. large-scale, non-symmetric and especially frequency-dependent. In order to ease this challenge, a three-step structure-preserving model order reduction method is presented, which is based on an offline-online computing framework. As a first step, the second-order Arnoldi process is used to speed up calculations of the sparse structural part of the strongly coupled system. In a second step, the low-dimensional approximations of the FE unknown variables are eliminated from the global system of equations and after that the remaining dense BE degrees of freedom are reduced by application of the standard Arnoldi-based Krylov subspace algorithm. A transformation technique based on Taylor’s theorem is incorporated to decouple the frequency from the Green’s function and a column-by-column low-memory projection is further sought to favor the overall storage requirements. In the last step, a robust reduced order model can be quickly retrieved by simple algebraic manipulations, which allows a direct solver without any costly matrix operation to be used for the online sweeps, requiring only a very small fraction of the total CPU runtime. Two numerical cases are investigated to highlight the potential of the proposed approach in multi-frequency applications.

A New Pseudolinear Filter for Bearings-Only Tracking without Requirement of Bias Compensation.

In bearings-only tracking systems, the pseudolinear Kalman filter (PLKF) has advantages in stability and computational complexity, but suffers from correlation problems. Existing solutions require bias compensation to reduce the correlation between the pseudomeasurement matrix and pseudolinear noise, but incomplete compensation may cause a loss of estimation accuracy. In this paper, a new pseudolinear filter is proposed under the minimum mean square error (MMSE) framework without requirement of bias compensation. The pseudolinear state-space model of bearings-only tracking is first developed. The correlation between the pseudomeasurement matrix and pseudolinear noise is thoroughly analyzed. By splitting the bearing noise term from the pseudomeasurement matrix and performing some algebraic manipulations, their cross-covariance can be calculated and incorporated into the filtering process to account for their effects on estimation. The target state estimation and its associated covariance can then be updated according to the MMSE update equation. The new pseudolinear filter has a stable performance and low computational complexity and handles the correlation problem implicitly under a unified MMSE framework, thus avoiding the severe bias problem of the PLKF. The posterior Cramer–Rao Lower Bound (PCRLB) for target state estimation is presented. Simulations are conducted to demonstrate the effectiveness of the proposed method.

Dynamics and Growth Rate Implications of Ribosomes and mRNAs Interaction in <i>E. coli</i>

Understanding how cells grow and adapt under various nutrient conditions is pivotal in the study of biological stoichiometry. Recent studies provide empirical evidence that cells use multiple strategies to maintain an optimal protein production rate under different nutrient conditions. Mathematical models can provide a solid theoretical foundation that can explain experimental observations and generate testable hypotheses to further our understanding of the growth process. In this study, we generalize a modeling framework that centers on the translation process and study its asymptotic behaviors to validate algebraic manipulations involving the steady states. Using experimental results on the growth of E. coli under C-, N-, and P-limited environments, we simulate the expected quantitative measurements to show the feasibility of using the model to explain empirical evidence. Our results support the findings that cells employ multiple strategies to maintain a similar protein production rate across different nutrient limitations. Moreover, we find that the previous study underestimates the significance of certain biological rates, such as the binding rate of ribosomes to mRNA and the transition rate between different ribosomal stages. Furthermore, our simulation shows that the strategies used by cells under C- and P-limitations result in a faster overall growth dynamics than under N-limitation. In conclusion, the general modeling framework provides a valuable platform to study cell growth under different nutrient supply conditions, which also allows straightforward extensions to the coupling of transcription, translation, and energetics to deepen our understanding of the growth process.

Interval linear quadratic regulator and its application for speed control of DC motor in the presence of uncertainties

An analytical investigation of a DC motor with interval uncertainties is performed in this study and a new approach by interval analysis is suggested for optimal control of the system. The main advantage of using an interval model for uncertainties is that makes the system independent from the probability distribution models of the system; therefore, it can be analyzed by only having information about minimum and maximum bounds. Here, the interval analysis deals with linear quadratic feedback control (LQR) to simulate and optimal control of the DC motor in the realistic state. To do this, the Pontryagins principle is used to solve the interval linear quadratic regulator to obtain the essential conditions, and thus, they have been reconstructed as ordinary differential equation by applying several algebraic manipulations. Afterward, by solving the interval nonlinear system of the ODE, the confidence interval for the feedback controller is achieved. The confidence interval is to guarantee the solution which is included in it. The Chebyshev inclusion approach is applied here to find solution for the ODE system with uncertainties. A comparison of the step response of the suggested approach with the centered approach and Monte Carlo methods a statistical approach is performed. The simulation results indicated that the suggested approach retains tighter and more sensible results than the Monte Carlo method.

Robust and Resource-Efficient Identification of Two Hidden Layer Neural Networks

We address the structure identification and the uniform approximation of two fully nonlinear layer neural networks of the type f(x)=1^T h(B^T g(A^T x)) on mathbb R^d, where g=(g_1,dots , g_{m_0}), h=(h_1,dots , h_{m_1}), A=(a_1|dots |a_{m_0}) in mathbb R^{d times m_0} and B=(b_1|dots |b_{m_1}) in mathbb R^{m_0 times m_1}, from a small number of query samples. The solution of the case of two hidden layers presented in this paper is crucial as it can be further generalized to deeper neural networks. We approach the problem by sampling actively finite difference approximations to Hessians of the network. Gathering several approximate Hessians allows reliably to approximate the matrix subspace mathcal W spanned by symmetric tensors a_1 otimes a_1,dots ,a_{m_0}otimes a_{m_0} formed by weights of the first layer together with the entangled symmetric tensors v_1 otimes v_1 ,dots ,v_{m_1}otimes v_{m_1}, formed by suitable combinations of the weights of the first and second layer as v_ell =A G_0 b_ell /Vert A G_0 b_ell Vert _2, ell in [m_1], for a diagonal matrix G_0 depending on the activation functions of the first layer. The identification of the 1-rank symmetric tensors within mathcal W is then performed by the solution of a robust nonlinear program, maximizing the spectral norm of the competitors constrained over the unit Frobenius sphere. We provide guarantees of stable recovery under a posteriori verifiable conditions. Once the 1-rank symmetric tensors {a_i otimes a_i, iin [m_0]}cup {v_ell otimes v_ell , ell in [m_1] } are computed, we address their correct attribution to the first or second layer (a_i’s are attributed to the first layer). The attribution to the layers is currently based on a semi-heuristic reasoning, but it shows clear potential of reliable execution. Having the correct attribution of the a_i,v_ell to the respective layers and the consequent de-parametrization of the network, by using a suitably adapted gradient descent iteration, it is possible to estimate, up to intrinsic symmetries, the shifts of the activations functions of the first layer and compute exactly the matrix G_0. Eventually, from the vectors v_ell =A G_0 b_ell /Vert A G_0 b_ell Vert _2’s and a_i’s one can disentangle the weights b_ell ’s, by simple algebraic manipulations. Our method of identification of the weights of the network is fully constructive, with quantifiable sample complexity and therefore contributes to dwindle the black box nature of the network training phase. We corroborate our theoretical results by extensive numerical experiments, which confirm the effectiveness and feasibility of the proposed algorithmic pipeline.

Nonlinear Deployment Dynamics and Wrinkling of a Membrane Attached to Two Axially Moving Beams

The out-of-plane and in-plane deployment dynamics of a flexible space structure, namely, a solar sail quadrant consisting of a membrane attached to two support booms, are considered. The equations of motion of the system are obtained using a time-varying generalization of the extended Hamilton’s principle. They are then discretized via quasi-modal expansion of the deflections as truncated series involving both time- and space-dependent basis functions. Because all directions are accounted for and plate strains are used to capture the potentially significant effect of even a small stiffness on the dynamics, nonlinear terms appear in the discretized equations. To increase computational efficiency, coordinate transformations and linear algebraic manipulations are performed to make all spatial integrals time invariant. In addition, attempts are made to predict wrinkling using the Miller–Hedgepeth model: a coarse mesh is defined, the instantaneous state of each region is determined using a wrinkling criterion and averaged principal stresses, and constitutive relation of each region is adjusted based on its wrinkling state. Numerical simulations provide basic validation, sample deployment results, and a comparison against the results of an earlier linear model with only out-of-plane deflections. The stress predictions are also partially validated using previous results based on constant-size loaded membrane experiments.

Optimal Perturbation Technique within the Asymptotic Iteration Method for Heavy-Light Meson Mass Splittings

For further insight into the perturbation technique within the framework of the asymptotic iteration method (PAIM), we suggest this method to be used as an alternative method to the traditional well-known perturbation techniques. We show by means of very simple algebraic manipulations that PAIM can be directly applied to obtain the symbolic expectation value of any perturbed potential piece without using the eigenfunction of the unperturbed problem. One of the fundamental advantages of PAIM is its ability to extract a reference unperturbed potential piece or pieces from the total Hamiltonian which can be solved exactly within AIM. After all, one can easily compute the symbolic expectation values of the remaining potential pieces. As an example, the present scheme is applied to the semi-relativistic wave equation with the harmonic-oscillator potential implemented with the Fermi–Breit potential terms. In particular, the non-trivial symbolic expectation values of the Dirac delta function, and the momentum-dependent orbit–orbit coupling terms are successfully calculated. Results are then used, as an illustration, to compute the semi-relativistic s-wave heavy-light meson masses. We obtain good agreement with experimental data for the meson mass splittings cu¯, cd¯, cs¯, bu¯, bd¯, bs¯.

Revisiting and generalizing the dual iteration for static and robust output‐feedback synthesis

AbstractThe dual iteration was introduced in a conference paper in 1997 by Iwasaki as an iterative and heuristic procedure for the challenging and non‐convex design of static output‐feedback controllers. We recall in detail its essential ingredients and go beyond the work of Iwasaki by demonstrating that the framework of linear fractional representations allows for a seamless extension of the dual iteration to output‐feedback designs of practical relevance, such as the design of robust or robust gain‐scheduled controllers. In the paper of Iwasaki, the dual iteration is solely based on, and motivated by algebraic manipulations resulting from the elimination lemma. We provide a novel control theoretic interpretation of the individual steps, which paves the way for further generalizations of the powerful scheme to situations where the elimination lemma is not applicable. As an illustration, we extend the dual iteration to a design of static output‐feedback controllers with multiple objectives. We demonstrate the approach with numerous numerical examples inspired from the literature.

Trajectory Tracking and Integrated Chassis Control for Obstacle Avoidance With Minimum Jerk

Intelligent vehicles are expected to perform emergency lane change maneuvers to avoid a collision. During this aggressive maneuver, a high jerk may occur, which reduces the comfort level and may be harmful to vehicle occupants. The present paper addresses autonomous collision avoidance in the context of minimum jerk. First, the desired trajectory described by the desired path and desired velocity profile is generated using quintic polynomials. These quintic polynomials are derived using the Euler-Lagrange equations for the functional defined as the time integral of the squared resultant jerk. The generation of the trajectory depends on the essential parameters, which are the initial longitudinal vehicle velocity, the desired final lateral position, and the tire-road friction coefficient. As a result of nondimensionalization and algebraic manipulations, the collision avoidance problem reduces to a nondimensionalized equation that is an implicit equation in one unknown, which is the lane change aspect ratio, and an input capturing the essential parameters. The plot of the lane change aspect ratio with respect to the input yields a curve that indicates the last point at which a collision can be avoided. The sliding mode control method is used to translate the trajectory tracking errors into the reference values of the total longitudinal force and the centers of percussion lateral accelerations that are the inputs to the tire force distributor. This distributor allocates these reference values to the tires by reducing the tire force usage. Numerical simulations at different initial longitudinal vehicle velocities demonstrate the effectiveness of the controller in avoiding the obstacle and keeping the vehicle motion stable.

Reference dependent invariant sets: Sum of squares based computation and applications in constrained control

The goal of this paper is to present a systematic method to compute reference dependent positively invariant sets for systems subject to constraints. To this end, we first characterize these sets as level sets of reference dependent Lyapunov functions. Based on this characterization and using Sum of Squares theory, we provide a polynomial certificate for the existence of such sets. Subsequently, through some algebraic manipulations, we express this certificate in terms of a Semi-Definite Programming problem which maximizes the size of the resulting reference dependent invariant sets. We then present some results implementing the proposed method to an example and propose some variants that may help in reducing possible numerical issues. Finally, the proposed approach is employed in the Model Predictive Control for Tracking scheme to compute the terminal set, and in the Explicit Reference Governor framework to compute the so-called Dynamic Safety Margin. The effectiveness of the proposed method in each of the schemes is demonstrated through simulation studies.

SuperTracer: a calculator of functional supertraces for one-loop EFT matching

We present SuperTracer, a Mathematica package aimed at facilitating the functional matching procedure for generic UV models. This package automates the most tedious parts of one-loop functional matching computations. Namely, the determination and evaluation of all relevant supertraces, including loop integration and Dirac algebra manipulations. The current version of SuperTracer also contains a limited set of output simplifications. However, a further reduction of the output to a minimal basis using Fierz identities, integration by parts, simplification of Dirac structures, and/or light field redefinitions might still be necessary. The code and example notebooks are publicly available at .1

Convex model predictive control for collision avoidance

This manuscript proposes a model predictive control for collision avoidance for the regulation problem of deterministic linear systems, which provides a priori guarantees of strong system theoretic properties, such as positive invariance and asymptotic stability, and high computational efficiency. Notion of safe distance sets is introduced, and also utilized as a novel approach to ensure collision avoidance via suitably defined convex constraints. The proposed convex model predictive control for collision avoidance is obtained by employing interactive strategic-tactical structure for overall decision-making. The strategic stage of the overall algorithm employs direct algebraic manipulations in order to construct safe distance sets that ensure collision avoidance. The tactical stage of the overall algorithm employs strictly convex quadratic programs for the optimization of local finite horizon predicted control processes. The dynamically compatible interaction of strategic and tactical stages of the overall algorithm is ensured by construction, which guarantees structural and computational benefits. These novel and unique features effectively enable both real time implementation and real life utilization of model predictive control for collision avoidance.

Elliptic tori in FPU non-linear chains with a small number of nodes

We revisit an algorithm constructing elliptic tori, that was originally designed for applications to planetary hamiltonian systems. The scheme is adapted to properly work with models of chains of N+1 particles interacting via anharmonic potentials, thus covering also the case of FPU chains. After having preliminarily settled the Hamiltonian in a suitable way, we perform a sequence of canonical transformations removing the undesired perturbative terms by an iterative procedure. This is done by using the Lie series approach, that is explicitly implemented in a programming code with the help of a software package, which is especially designed for computer algebra manipulations. In the cases of FPU chains with N=4,8, we successfully apply our new algorithm to the construction of elliptic tori for wide sets of the parameter ruling the size of the perturbation, i.e., the total energy of the system. Moreover, we explore the stability regions surrounding 1D elliptic tori. We compare our semi-analytical results with those provided by numerical explorations of the FPU-model dynamics, where the latter ones are obtained by using techniques based on the so called frequency analysis. We find that our procedure works up to values of the total energy that are of the same order of magnitude with respect to the maximal ones, for which elliptic tori are detected by numerical methods.

On tau-functions for the KdV hierarchy

For an arbitrary solution to the KdV hierarchy, the generating series of logarithmic derivatives of the tau-function of the solution can be expressed by the basic matrix resolvent via algebraic manipulations. Based on this we develop in this paper two new formulae for the generating series by introducing a pair of wave functions of the solution. Applications to the Witten–Kontsevich tau-function, to the generalized Brézin–Gross–Witten (BGW) tau-function, as well as to a modular deformation of the generalized BGW tau-function which we call the Lamé tau-function are also given.