Algebraic Manipulation Research Articles

Limit cycles of continuous and discontinuous piecewise-linear differential systems in R3

We study the limit cycles of two families of piecewise-linear differential systems in R 3 with two pieces separated by a plane Σ . In one family the differential systems are continuous on the plane Σ , and in the other family they are discontinuous on the plane Σ . The usual tool for studying these limit cycles is the Poincaré map, but here we shall use recent results which extend the averaging theory to continuous and discontinuous differential systems. All the computations have been done with the algebraic manipulator Mathematica.

Metric versus observable operator representation, higher spin models

.We elaborate further on the metric representation that is obtained by transferring the time-dependence from a Hermitian Hamiltonian to the metric operator in a related non-Hermitian system. We provide further insight into the procedure on how to employ the time-dependent Dyson relation and the quasi-Hermiticity relation to solve time-dependent Hermitian Hamiltonian systems. By solving both equations separately we argue here that it is in general easier to solve the former. We solve the mutually related time-dependent Schrödinger equation for a Hermitian and non-Hermitian spin 1/2, 1 and 3/2 model with time-independent and time-dependent metric, respectively. In all models the overdetermined coupled system of equations for the Dyson map can be decoupled algebraic manipulations and reduces to simple linear differential equations and an equation that can be converted into the non-linear Ermakov-Pinney equation.

Vocational High School Students Ability in Mathematics Literacy

Mathematical literacy is a person's ability to formulate, apply, and interpret mathematics in various contexts. This study is a qualitative descriptive that aims to describe the ability of mathematics literacy of students in thesubject of exponents and logarithms in Vocational High School. Data analysis is done by collecting data, reducing data, and verifying data. The results obtained in this study are subjects with values above the MEC cansolve the problem by using information, performing representations based on concepts, using procedural knowledge in the form of algebraic manipulation in accordance with the nature of exponents and logarithms,and can connect it with the real world to determine the outcome of completion, according to level 4 PISA. Subjects with the same value as MEC can solve problems, interpret by using the properties of exponents andlogarithms, and implement procedures according to the 3rd level of PISA. Subjects below the MEC can only solve commonly resolved problems using simple properties, corresponding to the PISA 1st level.

Construction and nonexistence of strong external difference families

Strong external difference families (SEDFs) were introduced by Paterson and Stinson as a more restrictive version of external difference families. SEDFs can be used to produce optimal strong algebraic manipulation detection codes. We characterize the parameters $(v, m, k, \lambda)$ of a nontrivial SEDF that is near-complete (satisfying $v=km+1$). We construct the first known nontrivial example of a $(v, m, k, \lambda)$ SEDF having $m > 2$. The parameters of this example are $(243,11,22,20)$, giving a near-complete SEDF, and its group is $\mathbb{Z}_3^5$. We provide a comprehensive framework for the study of SEDFs using character theory and algebraic number theory, showing that the cases $m=2$ and $m>2$ are fundamentally different. We prove a range of nonexistence results, greatly narrowing the scope of possible parameters of SEDFs.

Prospective Middle-School Mathematics Teachers’ Quantitative Reasoning and Their Support for Students’ Quantitative Reasoning

The aim of this research is to examine prospective mathematics teachers’ quantitative reasoning, their support for students’ quantitative reasoning and the relationship between them, if any. The teaching experiment was used as the research method in this qualitatively designed study. The data of the study were collected through a series of exploratory teaching interviews and debriefing interviews with nine focus group participants, and clinical interviews that the participants conducted with middle-school students. The results indicated that the participants with strong quantitative reasoning use problem-solving approaches that focused on the quantity, whereas the participants with poor quantitative reasoning use problem-solving approaches that focused on performing calculations, using formulas and procedures devoid of quantitative meaning in solving of the problem. During the questioning process, the participants with strong quantitative reasoning led their students to identify and interpret the quantities, determine relationships among the quantities, represent all the quantities and their interrelationships, whereas the participants with poor quantitative reasoning led their students to perform calculations, make algebraic manipulations and focus on numbers by ignoring the quantities in the problem. These results suggest that prospective mathematics teachers’ quantitative reasoning is strongly associated with their support for students’ quantitative reasoning in the problem-solving process.

Generalized Kleinman‐Newton method

SummaryThis paper addresses the general problem of optimal linear control design subject to convex gain constraints. Classical approaches based exclusively on Riccati equations or linear matrix inequalities are unable to treat problems that incorporate feedback gain constraints, for instance, the reduced‐order (including static) output feedback control design. In this paper, these two approaches are put together to obtain a genuine generalization of the celebrated Kleinman‐Newton method. The convergence to a local minimum is monotone. We believe that other control design problems can be also considered by the adoption of the same ideas and algebraic manipulations. Several examples borrowed from the literature are solved for illustration and comparison.

Analysis of coined quantum walks with renormalization

We introduce a new framework to analyze quantum algorithms with the renormalization group (RG). To this end, we present a detailed analysis of the real-space RG for discrete-time quantum walks on fractal networks and show how deep insights into the analytic structure as well as generic results about the long-time behavior can be extracted. The RG-flow for such a walk on a dual Sierpinski gasket and a Migdal-Kadanoff hierarchical network is obtained explicitly from elementary algebraic manipulations, after transforming the unitary evolution equation into Laplace space. Unlike for classical random walks, we find that the long-time asymptotics for the quantum walk requires consideration of a diverging number of Laplace-poles, which we demonstrate exactly for the closed form solution available for the walk on a 1d-loop. In particular, we calculate the probability of the walk to overlap with its starting position, which oscillates with a period that scales as $N^{d_{w}^{Q}/d_{f}}$ with system size $N$. While the largest Jacobian eigenvalue $\lambda_{1}$ of the RG-flow merely reproduces the fractal dimension, $d_{f}=\log_{2}\lambda_{1}$, the asymptotic analysis shows that the second Jacobian eigenvalue $\lambda_{2}$ becomes essential to determine the dimension of the quantum walk via $d_{w}^{Q}=\log_{2}\sqrt{\lambda_{1}\lambda_{2}}$. We trace this fact to delicate cancellations caused by unitarity. We obtain identical relations for other networks, although the details of the RG-analysis may exhibit surprisingly distinct features. Thus, our conclusions -- which trivially reproduce those for regular lattices with translational invariance with $d_{f}=d$ and $d_{w}^{Q}=1$ -- appear to be quite general and likely apply to networks beyond those studied here.

Local null controllability of the control-affine nonlinear systems with time-varying disturbances

The problem of local null controllability for the control-affine nonlinear systems x˙(t)=f(x(t))+Bu(t)+w(t),t ∈ [0, T] is considered in this paper. The principal requirements on the system are that the LTI pair ((∂f/∂x)(0), B) is controllable and the disturbance is limited by the constraint |f(0)+w(t)|≤Md(1−tT)η,Md ≥ 0 and η > 0. These properties together with one technical assumption yield an answer to the problem of deciding when the null controllable region has a nonempty interior. The obtained criterion is built on the purely algebraic and/or differential manipulations with vector field f, input matrix B and a bound on the disturbance w(t). To prove the main result we have derived a new Gronwall-type inequality allowing the fine estimates of the closed-loop solutions. The theory is illustrated and the efficacy of proposed controller is demonstrated by the example where the null controllable region is explicitly calculated. Finally, we established the sufficient conditions to be the system under consideration with w(t) ≡ 0 globally null controllable.

Relay Selection and Resource Assignment for Cooperative SC-FDMA Networks

In this article, we study relay selection and resource allocation in a cooperative network that employs Single Carrier - Frequency Division Multiple Access (SC-FDMA) in the uplink direction such as in Long Term Evolution - Advanced (LTE-A) system. With SC-FDMA, the frequency resources assigned to a transmitter should be adjacent to each other in the frequency domain (adjacency constraint). We formulate the total data rate maximization problem as a non-linear combinatorial optimization problem. Through algebraic manipulations, we managed to convert the original problem in a Mixed Integer Linear Problem (MILP) that can be optimally solved by standard computational solvers. So as to reduce the computational complexity, we propose a low-complexity suboptimal solution to this problem. Additionally, we propose a transmit power adjustment in order to reduce the power consumption in the system and improve Energy Efficiency (EE) by exploiting the characteristics of the forwarding mechanism in the relays. Finally, we present simulation results so as to evaluate the performance of the proposed solutions and obtain insights on the effect of the main variables presented in the system model.

A Constrained-Total-Least-Squares Method for Joint Estimation of Source and Sensor Locations: A General Framework

It is well known that sensor location uncertainties can significantly deteriorate the source positioning accuracy. Therefore, improving the sensor locations is necessary in order to achieve better localization performance. In this paper, a constrained-total-least-squares (CTLS) method for simultaneously locating multiple disjoint sources and refining the erroneous sensor positions is presented. Unlike conventional localization techniques, an important feature of the proposed method is that it establishes a general framework that is suitable for many different location measurements. First, a modified CTLS optimization problem is formulated after some algebraic manipulations and then the corresponding Newton iterative algorithm is derived to give the numerical solution. Subsequently, by using the first-order perturbation analysis, the explicit expression for the covariance matrix of the proposed CTLS estimator is deduced under the Gaussian assumption. Moreover, the estimation accuracy of the CTLS method is shown to achieve the Cramér-Rao bound (CRB) before the thresholding effect occurs by a rigorous proof. Finally, two kinds of numerical examples are given to corroborate the theoretical development in this paper. One uses the TDOAs/GROAs measurements and the other is based on the TOAs/FOAs parameters.

The Linear and Quadratic Nature of Sequences for Determining Prime Numbers by Elimination of Composites

A previously derived set of sequences for generating prime numbers is modified by various algebraic and trigonometric manipulations. The modified sequences are then used to more efficiently generate prime numbers. The modified sequences are simplified from their original form and are much easier to implement into a computer program. It is shown by computation that the modified sequences more efficiently determine prime numbers because the modified sequences provide flexibility with their implementation. The added flexibility allows for the removal of redundancy in the overall process, which decreases the amount of data to process; thus, the efficiency of the prime number generator is significantly improved.

Classification of Bent Monomials, Constructions of Bent Multinomials and Upper Bounds on the Nonlinearity of Vectorial Functions

This paper is composed of two main parts related to the nonlinearity of vectorial functions. The first part is devoted to maximally nonlinear $(n,m)$ functions (the so-called bent vectorial functions), which contribute to an optimal resistance to both linear and differential attacks on symmetric cryptosystems. They can be used in block ciphers at the cost of additional diffusion/compression/expansion layers, or as building blocks for the construction of substitution boxes (S-boxes), and they are also useful for constructing robust codes and algebraic manipulation detection codes. A main issue on bent vectorial functions is to characterize bent monomial functions $Tr_{m}^{n} (\lambda x^{d})$ from $\mathbb {F}_{2^{n}}$ to $\mathbb {F}_{2^{m}}$ (where $m$ is a divisor of $n$ ) leading to a classification of those bent monomials. We also treat the case of functions with multiple trace terms involving general results and explicit constructions. Furthermore, we investigate some open problems raised by Pasalic et al. and Muratovic-Ribic et al. in a series of papers on vectorial functions. The second part is devoted to the nonlinearity of $(n,m)$ -functions. No tight upper bound is known when $\frac {n}2 . The covering radius bound is the only known upper bound in this range (the Sidelnikov–Chabaud–Vaudenay bound coincides with it when $m=n-1$ and it has no sense when $m ). Finding better bounds is an open problem since the 1990s. Moreover, no bound has been found during the last 23 years, which improve upon the covering radius bound for a large part of $(n,m)$ -functions. We derive such upper bounds for functions, which are sufficiently unbalanced or which satisfy some conditions. These upper bounds imply some necessary conditions for vectorial functions to have large nonlinearity.

A new method based on artificial neural networks for solving general nonlinear systems

Implementation of the amazing features of the human brain in an artificial system has long been considered. It seems that simulating the human nervous system is a recent development in applied mathematics and computer sciences. The objective of this research is to introduce an efficient iterative method based on artificial neural networks for numerically solving nonlinear algebraic systems of polynomial equations. The method first performs some simple algebraic manipulations to convert the origin system to an approximated unconstrained optimisation problem. Subsequently, the resulting nonlinear minimisation problem is solved iteratively using the neural networks approach. For this aim, a suitable five-layer feed-back neural architecture is formed and trained using a back-propagation supervised learning algorithm which is based on the gradient descent rule. Ultimately, some numerical examples with comparisons are given to demonstrate the high accuracy and the ease of implementation of the present technique over other classical methods.

A new method based on artificial neural networks for solving general nonlinear systems

Implementation of the amazing features of the human brain in an artificial system has long been considered. It seems that simulating the human nervous system is a recent development in applied mathematics and computer sciences. The objective of this research is to introduce an efficient iterative method based on artificial neural networks for numerically solving nonlinear algebraic systems of polynomial equations. The method first performs some simple algebraic manipulations to convert the origin system to an approximated unconstrained optimisation problem. Subsequently, the resulting nonlinear minimisation problem is solved iteratively using the neural networks approach. For this aim, a suitable five-layer feed-back neural architecture is formed and trained using a back-propagation supervised learning algorithm which is based on the gradient descent rule. Ultimately, some numerical examples with comparisons are given to demonstrate the high accuracy and the ease of implementation of the present technique over other classical methods.

Direct Realization of Digital Differentiators in Discrete Domain for Active Damping of LCL -Type Grid-Connected Inverter

To damp the LCL-filter resonance in a grid-connected inverter, the feedback of capacitor current is usually adopted, and it can be replaced by the feedback of capacitor voltage as a low-cost solution, if an accurate digital differentiator can be made. The best way for realizing such a differentiator has so far proved to be an indirect nonideal generalized integrator (GI). As a simple alternative, this paper proposes two digital differentiators, which are directly developed in the discrete domain. They are a first-order differentiator based on backward Euler plus digital lead compensator and a second-order differentiator based on Tustin plus digital notch filter. The basic idea of the proposed methods is to correct their frequency responses to match the ideal differentiator with embedded digital filters. It is shown that the proposed differentiators exhibit the same derivative performance as the nonideal-GI differentiator, and they are more attractive for digital implementations due to their direct discrete natures, compact expressions, and easy algebraic manipulations. In particular, the proposed first-order differentiator is most competitive for its general representation and simplest implementation. Finally, a 12-kW prototype is built, and experiments are performed to verify the theoretical analysis.

A method of providing the integrity of information in the group of robotic engineering complexes based on crypt-code constructions

The system of cryptographic code transformation of information based on the aggregated application of block encryption algorithms and polynomial codes of the residual class system has been considered. The complexation of information processing methods ensures the restoration of the integrity of information stored in a group of robotic complexes (RCs) that is subjected to the actions (algebraic manipulations) of the violator, while the physical loss of some predetermined limit number of RCs does not lead to the partial or complete loss of it.

Algebraic manipulation detection codes with perfect nonlinear functions under non-uniform distribution

Algebraic manipulation detection codes with perfect nonlinear functions under non-uniform distribution

Efficient and accurate numerical simulation of acoustic wave propagation in a 2D heterogeneous media

In this paper, a compact fourth-order finite difference scheme is derived to solve the 2D acoustic wave equation in heterogenous media. The Padé approximation is used to obtain fourth-order accuracy in both temporal and spatial dimensions, and the alternating direction implicit (ADI) technique is used to reduce the computational cost. Due to the non-constant wave velocity, the conventional ADI method is hard to implement as the algebraic manipulation cannot be used here. A novel numerical strategy is proposed in this work so that the compact scheme still maintains fourth-order accuracy in time and space. The fourth-order convergence order was firstly proved by theoretical error analysis, then was confirmed by numerical examples. It was shown that the proposed method is conditionally stable with a Courant–Friedrichs–Lewy (CFL) condition that is comparable to other existing finite difference schemes. Several numerical examples were solved to demonstrate the efficiency and accuracy of the new algorithm.

Automated calculation of matrix elements and physics motivated observables

The central aspect of my personal scientific activity, has focused on calculations useful for interpretation of High Energy accelerator experimental results, especially in a domain of precision tests of the Standard Model. My activities started in early 80’s, when computer support for algebraic manipulations was in its infancy. But already then it was important for my work. It brought a multitude of benefits, but at the price of some inconvenience for physics intuition. Calculations became more complex, work had to be distributed over teams of researchers and due to automatization, some aspects of the intermediate results became more difficult to identify.In my talk I will not be very exhaustive, I will present examples from my personal research only: (i) calculations of spin effects for the process e+e− → τ+τ−γ at Petra/PEP energies, calculations (with the help of the Grace system of Minami-tateya group) and phenomenology of spin amplitudes for (ii) e+e− → 4f and for (iii) e+e− → νeν̄eγγ processes, (iv) phenomenology of CP-sensitive observables for Higgs boson parity in H → τ+τ−, τ± → ν2(3)π cascade decays.

A new algorithm for computing branching rules and Clebsch–Gordan coefficients of unitary representations of compact groups

A numerical algorithm that computes the decomposition of any finite-dimensional unitary reducible representation of a compact Lie group is presented. The algorithm, which does not rely on an algebraic insight into the group structure, is inspired by quantum mechanical notions. After generating two adapted states (these objects will be conveniently defined in Definition II.1) and after appropriate algebraic manipulations, the algorithm returns the block matrix structure of the representation in terms of its irreducible components. It also provides an adapted orthonormal basis. The algorithm can be used to compute the Clebsch–Gordan coefficients of the tensor product of irreducible representations of a given compact Lie group. The performance of the algorithm is tested on various examples: the decomposition of the regular representation of two finite groups and the computation of Clebsch–Gordan coefficients of two examples of tensor products of representations of SU(2).