Polynomial Problems Research Articles

MATHEMA TIC-TAC-TOE: AN INTERVENTION TO ENHANCE POLYNOMIAL-SOLVING SKILLS AMONG STUDENTS IN KAPALONG NATIONAL HIGH SCHOOL

The purpose of this study is to evaluate the impact of the Mathema Tic-Tac-Toe intervention on students’ skills in solving polynomial problems. Specifically, the study aims to determine the significant difference between students’ pretest and post-test scores. A pre-experimental research design was utilized to assess the intervention’s effectiveness. Data were collected from 76 students in Grades 8 and 10 at Kapalong National High School, selected through purposive sampling. Pre-test and post-test assessments were used for data gathering. The findings revealed a substantial improvement in scores, with the mean percentage score increasing from 15.5% (Very Low) before the intervention to 61.14% (Average) after the intervention. The t-value obtained was -22.19, indicating a substantial difference in the scores before and after the intervention or treatment related to solving polynomials. This large negative t-value suggests a strong effect of the treatment on the students’ ability to solve polynomials. This significant improvement in scores indicates a highly positive impact of the Mathema Tic-Tac-Toe intervention on students’ ability to solve polynomial problems. The results suggest that the intervention is effective in enhancing students’ performance in this area, demonstrating a strong and beneficial effect on their Mathematical skills. The teacher may use this intervention in their mathematics classes to improve the problem solving skills of the students in polynomials. KEYWORDS: Action Research, Intervention, Polynomial Problem Solving Skills, Mathema Tic-Tac-Toe

Hierarchy relaxations for robust equilibrium constrained polynomial problems and applications to electric vehicle charging scheduling

In this paper, we consider a polynomial problem with equilibrium constraints in which the constraint functions and the equilibrium constraints involve data uncertainties. Employing a robust optimization approach, we examine the uncertain equilibrium constrained polynomial optimization problem by establishing lower bound approximations and asymptotic convergences of bounded degree diagonally dominant sum-of-squares (DSOS), scaled diagonally dominant sum-of-squares (SDSOS) and sum-of-squares (SOS) polynomial relaxations for the robust equilibrium constrained polynomial optimization problem. We also provide numerical examples to illustrate how the optimal value of a robust equilibrium constrained problem can be calculated by solving associated relaxation problems. Furthermore, an application to electric vehicle charging scheduling problems under uncertain discharging supplies shows that for the lower relaxation degrees, the DSOS, SDSOS and SOS relaxations obtain reasonable charging costs and for the higher relaxation degrees, the SDSOS relaxation scheme has the best performance, making it desirable for practical applications.

Exploring the Potential of DNA Computing for Complex Big Data Problems: A Case Study on the Traveling Car Renter Problem.

The traveling car renter problem (TCRP) is a variant of the Traveling Salesman Problem (TSP) wherein the salesman utilizes rented cars for travel. The primary objective of this problem is to identify a solution that minimizes the cumulative operating costs. Given its classification as a non-deterministic polynomial (NP) problem, traditional computers are not proficient in effectively resolving it. Conversely, DNA computing exhibits unparalleled advantages when confronted with NP-hard problems. This paper presents a DNA algorithm, based on the Adleman-Lipton model, as a proposed approach to address TCRP. The solution for TCRP can be acquired by following a series of fundamental steps, including coding, interaction, and extraction. The time computing complexity of the proposed DNA algorithm is O(n2m) for TCRP with n cities and m types of cars. By conducting simulation experiments, the solutions for certain instances of TCRP are computed and compared to those obtained by alternative algorithms. The proposed algorithm further illustrates the potential of DNA computing, as a form of parallel computing, to address more intricate large-scale problems.

Cryptographic Protocols Resilient to Quantum Attacks: Advancements in Post-Quantum Cryptography

New developments in quantum computing have made people worry that current security methods could be broken by quantum attacks. Some mathematical problems are hard to solve, which is how traditional encryption methods like RSA and ECC keep their secrets safe. However, quantum computers might be able to solve these issues a lot faster than regular computers, which makes these methods less safe. Because of this threat, experts have been working hard to make post-quantum cryptography methods that quantum computers can't break. These methods are made to stand up to the strength of quantum algorithms and protect the privacy, integrity, and validity of private data even when quantum attackers are present. Lattice-based cryptography is a potential method for post-quantum encryption. Its security comes from the difficulty of certain lattice problems. Lattice-based methods are thought to be safe from threats from both traditional and quantum computers because they offer good security promises. Lattice-based cryptography is a flexible way to build different types of cryptographic primitives, like encryption, digital signatures, and key sharing protocols. The study of code-based cryptography, which is based on how hard it is to decode some error-correcting codes, is another important advance. Code-based methods have been around for a long time and have been studied a lot, which makes them a good choice for security after quantum computing. Code-based cryptography is also easy to use and has strong security features, which makes it a good choice for real-world situations. Multivariate polynomial cryptography is another possible choice for cryptography after quantum computing. The safety of this method depends on how hard it is to solve systems of multivariate polynomial problems. Multivariate polynomial methods might not be as secure as lattice-based or code-based cryptography, but they are interesting options for some situations and places with limited resources.

Berezin density and planar orthogonal polynomials

We introduce a nonlinear potential theory problem for the Laplacian, the solution of which characterizes the Berezin density B ( z , ⋅ ) B(z,\cdot ) for the polynomial Bergman space, where the point z ∈ C z\in \mathbb {C} is fixed. When z = ∞ z=\infty , the Berezin density is expressed in terms of the squared modulus of the corresponding normalized orthogonal polynomial P P . We use an approximate version of this characterization to study the asymptotics of the orthogonal polynomials in the context of exponentially varying weights. This builds on earlier works by Its-Takhtajan and by the first author on a soft Riemann-Hilbert problem for planar orthogonal polynomials, where in place of the Laplacian we have the ∂ ¯ \bar \partial -operator. We adapt the soft Riemann-Hilbert approach to the nonlinear potential problem, where the nonlinearity is due to the appearance of | P | 2 |P|^2 in place of P ¯ \overline {P} . Moreover, we suggest how to adapt the potential theory method to the study of the asymptotics of more general Berezin densities B ( z , w ) B(z,w) in the off-spectral regime, that is, when z z is fixed outside the droplet. This is a first installment in a program to obtain an explicit global expansion formula for the polynomial Bergman kernel, and, in particular, of the one-point function of the associated random normal matrix ensemble.

Dynamic resource allocation for energy-efficient downlink NOMA systems in 5G networks

Non-Orthogonal Multiple Access (NOMA) is a promising energy-efficient technology designed to satisfy the demands of future networks by efficiently sharing radio resources. In NOMA, the same radio resource is simultaneously assigned to two users at different power levels based on the NOMA-power domain. Resource allocation in NOMA presents a non-convex challenge, characterized as a non-deterministic polynomial (NP-hard) problem. This involves user and channel assignment and power allocation, making it an extraordinarily complex task to achieve an optimal solution. In this work, Simulated Annealing (SA) is proposed as an optimization technique for resource allocation in an energy-efficient downlink NOMA system. This resource allocation scheme addresses user and channel assignment, as well as power allocation, using SA as an efficient standalone approach to maximize energy efficiency in NOMA. SA is utilized to execute the assignment of users to channels, distribute the necessary power for each channel, and determine the power ratio among users sharing the same channel. The results obtained demonstrate a significant improvement in energy efficiency, outperforming the existing numerical methods by 22 %. The proposed SA scheme not only achieves a close optimal solution but also in less computational time, offering sufficient reliability in terms of energy efficiency enhancement when compared to the existing numerical method.

Kalman filtering and sine arithmetic optimization algorithm (KSAOA) for wireless sensor network clustering

Hierarchical routing is a crucial approach for enhancing the energy efficiency of Wireless Sensor Networks (WSN). The selection of sensor clusters poses a Non-deterministic Polynomial (NP) problem, typically addressed using meta-heuristic algorithms. However, existing research has not adequately considered the observation error of node energy and the limited convergence performance of their meta-heuristic algorithms. To address these challenges, we proposes a novel approach called KSAOA (Kalman filtering and Sine Arithmetic Optimization Algorithm) for WSN clustering, aimed at prolonging the service lifetime and enhancing the network balance of WSN. The proposed model combines Kalman filtering for estimating node energy and the Sine Arithmetic Optimization Algorithm (SAOA) for clustering nodes. SAOA builds upon the Arithmetic Optimization algorithm (AOA) by introducing a sine factor to extend the exploration boundary and a levy flight strategy to improve local exploration. Additionally, greedy algorithms are employed in the data transfer phase to enhance operational efficiency. The experimental results demonstrate the effectiveness of the proposed method in tabular and graphical form. The results are analyzed through comparative experiments to show that the proposed strategy is effective in improving the balance of network lifetime and network energy consumption.

Policy iteration for H∞ control of polynomial time‐varying systems

AbstractThis paper studies the H∞ control problem for polynomial time‐varying systems. The H∞ control problem has been much less investigated for time‐varying systems in comparison to the time‐invariant systems. Approximate dynamic programming (ADP) is an optimal method to solve the control problems. Therefore, it is valuable to solve the polynomial time‐varying H∞ control problem with the ADP approach. Considering the time as an independent variable for sum‐of‐squares (SOS) optimization problems, an SOS‐based ADP method is proposed to solve this problem. A policy iteration algorithm is presented, where in its policy evaluation step it is sufficient to solve an optimization problem. Some constraints are added to this optimization problem to guarantee the closed‐loop exponential stability. The convergence and stability properties of the proposed algorithm are stated and proven. Moreover, in order to design an H∞ controller with a smaller disturbance attenuation coefficient, a two‐loop algorithm is suggested. Finally, the effectiveness of the proposed method is demonstrated by simulation examples.

Hardness of Random Optimization Problems for Boolean Circuits, Low-Degree Polynomials, and Langevin Dynamics

Hardness of Random Optimization Problems for Boolean Circuits, Low-Degree Polynomials, and Langevin Dynamics

On the power sums problem of bi-periodic Fibonacci and Lucas polynomials

<abstract><p>This paper mainly discussed the power sums of bi-periodic Fibonacci and Lucas polynomials. In addition, we generalized these results to obtain several congruences involving the divisible properties of bi-periodic Fibonacci and Lucas polynomials.</p></abstract>

An optimal approximation problem for free polynomials

Motivated by recent work on optimal approximation by polynomials in the unit disk, we consider the following noncommutative approximation problem: for a polynomial f f in d d freely noncommuting arguments, find a free polynomial p n p_n , of degree at most n n , to minimize c n ≔ ‖ p n f − 1 ‖ 2 c_n ≔\|p_nf-1\|^2 . (Here the norm is the ℓ 2 \ell ^2 norm on coefficients.) We show that c n → 0 c_n\to 0 if and only if f f is nonsingular in a certain nc domain (the row ball), and prove quantitative bounds. As an application, we obtain a new proof of the characterization of polynomials cyclic for the d d -shift.

Quality of service aware improved coati optimization algorithm for efficient task scheduling in cloud computing environment

The task scheduling has a potential impact on overall system performance and resource utilization in the domain of Cloud Computing. Cloud computing adopts a cloud service provider (CSP) for facilitating access and delivering services to the shared resources. Task scheduling in clouds can be attained some symmetry form which helps in achieving predominant resource optimization that includes energy efficiency and load balancing. This task scheduling process of cloud computing pertains to the problem of Non-deterministic Polynomial (NP) which can be significantly solved using the techniques of metaheuristic optimization for enhancing the job scheduling effectiveness. In this paper, an Improved Coati Optimization Algorithm-based Task Scheduling (ICOATS) is presented for addressing the issues of lengthier scheduling time, high consumptions of cost and maximized load on Virtual Machine (VM) in cloud computing environment. In this proposed ICOATS, a model for distribution and scheduling of tasks is constructed using the factors of VMs, cost and time. It further included a multi-objective fitness function which targets on minimizing makespan, and at the same time maximizing the rate of resource utilization. It established possible plan for every coati with respect to task scheduling process that aids in determining the best solution (optimal assignment of incoming tasks to VMs). It is proposed with the capability of handing the problem of premature convergence by incorporating an exploitation strategy which improving the local search potential with well-balanced trade-off amid exploration and exploitation. The simulation results of this ICOATS approach under its evaluation with the existing metaheuristic task scheduling approaches confirmed better improvement in reducing makespan, and simultaneously enhances the turnaround efficiency, success rate and availability.

Conditional GrÖbner Basis: GrÖbner Basis Detection with Parameters

For a given finite polynomial set, finding a monomial order such that the given set is already a GrÖbner basis for the ideal generated by the given set with respect to the found monomial order is called GrÖbner basis detection (GBD) problem and there is also its simpler version, called structural GrÖbner basis detection (SGBD) problem. In this short communication, we give algorithms to solve these problems for polynomials with parameters on their coefficients.

A Hybrid Many-Objective Optimization Algorithm for Job Scheduling in Cloud Computing Based on Merge-and-Split Theory

Scheduling jobs within a cloud environment is a critical area of research that necessitates meticulous analysis. It entails the challenge of optimally assigning jobs to various cloud servers, each with different capabilities, and is classified as a non-deterministic polynomial (NP) problem. Many conventional methods have been suggested to tackle this difficulty, but they often struggle to find nearly perfect solutions within a reasonable timeframe. As a result, researchers have turned to evolutionary algorithms to tackle this problem. However, relying on a single metaheuristic approach can be problematic as it may become trapped in local optima, resulting in slow convergence. Therefore, combining different metaheuristic strategies to improve the overall system enactment is essential. This paper presents a novel approach that integrates three methods to enhance exploration and exploitation, increasing search process efficiency and optimizing many-objective functions. In the initial phase, we adopt cooperative game theory with merge-and-split techniques to train computing hosts at different utilization load levels, determining the ideal utilization for each server. This approach ensures that servers operate at their highest utilization range, maximizing their profitability. In the second stage, we incorporate the mean variation of the grey wolf optimization algorithm, making significant adjustments to the encircling and hunting phases to enhance the exploitation of the search space. In the final phase, we introduce an innovative pollination operator inspired by the sunflower optimization algorithm to enrich the exploration of the search domain. By skillfully balancing exploration and exploitation, we effectively address many-objective optimization problems. To validate the performance of our proposed method, we conducted experiments using both real-world and synthesized datasets, employing CloudSim software version 5.0. The evaluation involved two sets of experiments to measure different evaluation metrics. In the first experiment, we focused on minimizing factors such as energy costs, completion time, latency, and SLA violations. The second experiment, in contrast, aimed at maximizing metrics such as service quality, bandwidth utilization, asset utilization ratio, and service provider outcomes. The results from these experiments unequivocally demonstrate the outstanding performance of our algorithm, surpassing existing state-of-the-art approaches.

THSI-RP: A two-tier hybrid swarm intelligence based node clustering and multi-hop routing protocol optimization for wireless sensor networks

Wireless sensor networks exploit clustering and routing techniques to improve energy efficiency, but these methods are generally considered as a non-deterministic polynomial (NP-hard) problem. To tackle this problem, we propose a novel two-tier hybrid swarm intelligence-based hierarchical routing protocol (THSI-RP). In the first tier, THIS-RP incorporates a hybrid swarm intelligence algorithm that combines Grey Wolf Optimization (GWO) and Marine Predators Algorithm (MPA) for the clustering algorithm. By incorporating the wolf factors from GWO into MPA as elite predators, we enhance the network search efficiency. This algorithm achieves adaptive optimal clustering with a controllable scale, considering the residual energy of nodes, relative distance between nodes, and node centrality. In the second tier, the routing algorithm uses a hybrid SI algorithm based on GWO and the graph model. First, we integrate distance and energy factors to dynamically select forwarding nodes using GWO. Then, based on the distance and energy balance principle, a weight cost function is established and combined with the minimum spanning tree method to construct a communication routing tree between the forwarding node and the base station to achieve optimized inter-cluster multi-hop routing. Simulation results demonstrate that THSI-RP outperforms several typical routing protocols in various metrics over EEUC, LPSO, LGWO, and LACO. For instance, THSI-RP achieves percentage improvement total packets by 290%, 10.9%, 8.9%, 19.9%, respectively. It also performs better in FND by 88.7%, 10.9%, 8.6%, and 16.3%, in HDN by 86.6%, 8.1%, 6.3%, and 17.1%, and in ADN by 41.7%, 8.2%, 6.4%, 18.3%.

On the Koebe quarter theorem for trinomials with fold symmetry

The Koebe problem for univalent polynomials with real coefficients is fully solved only for trinomials, which means that in this case the Koebe radius and the extremal polynomial (extremizer) have been found. The general case remains open, but conjectures have been formulated. The corresponding conjectures have also been hypothesized for univalent polynomials with real coefficients and T T -fold rotational symmetry. This paper provides confirmation of these hypotheses for trinomials z + a z T + 1 + b z 2 T + 1 z + az^{T + 1} + bz^{2T + 1} . Namely, the Koebe radius is r = 4 cos 2 ⁡ ( π ( 1 + T ) 2 + 3 T ) r=4\cos ^2\left (\frac {\pi (1+T)}{2+3T}\right ) , and the only extremizer of the Koebe problem is the trinomial B ( T ) ( z ) = z + 2 2 + 3 T ( − T + ( 2 + 2 T ) cos ⁡ ( π T 2 + 3 T ) ) z 1 + T + + 1 2 + 3 T ( 2 + T − 2 T cos ⁡ ( π T 2 + 3 T ) ) z 1 + 2 T . \begin{gather*} B^{(T)}(z)=z+\frac 2{2+3T}\left (-T+(2+2T)\cos \left (\frac {\pi T}{2+3T}\right )\right )z^{1+T}+\\ +\frac 1{2+3T}\left (2+T-2T\cos \left (\frac {\pi T}{2+3T}\right )\right )z^{1+2T}. \end{gather*}

Bojanov–Naidenov Problem for Differentiable Functions and the Erdős Problem for Polynomials and Splines

Bojanov–Naidenov Problem for Differentiable Functions and the Erdős Problem for Polynomials and Splines

An Extremal Problem for Odd Univalent Polynomials

An Extremal Problem for Odd Univalent Polynomials

A two-dimensional wave and Rayleigh–Ritz method for complex dispersion in periodic arrays of circular damped acoustic black holes

Two-dimensional arrays of circular acoustic black holes (ABHs) on plates are known to exhibit many surprising properties such as bandgap formation, scattering, topological edge states and negative refraction and bi-refraction, to name a few. These effects can be deduced from the analysis of dispersion relations. However, to date, most of the work on periodic ABH arrays has only dealt with real dispersion curves that do not adequately reproduce their behavior. The role of the damping layer and intrinsic losses, which is essential to ABHs, is ignored. This can be accounted for by complex dispersion. In this paper, we extend the wave and Rayleigh–Ritz method (WRRM) previously developed for periodic ABHs in beams to address complex dispersion in two-dimensional periodic ABH arrays. This poses several challenges. On the one hand, in the WRRM the solution of the equations of motion is expanded in terms of the basis of the nullspace determined by the periodic boundary conditions. Although this only involves point connections for beams, in the two-dimensional case continuous line constraints are present and care must be taken to find the appropriate nullspace. To obtain the complex dispersion surface and complex equi-frequency contours of two-dimensional periodic ABH arrays we then need to set the eigenvalue problem in the k(ω) approach. For a fixed angular frequency ω∈R and a direction of propagation, θ, whose tangent is the quotient of the unknown wavenumber components in the y and x directions, we derive polynomial and transcendental eigenvalue problems for the complex wavenumber k∈ℂ×ℂ as a function of θ. The complex dispersion surface and equi-frequency contours of the periodic ABH arrays can be obtained by solving these eigenvalue problems with an iteration method. The two-dimensional WRRM is validated by comparison with finite element simulations. Finally, we apply it to calculate the complex dispersion of an infinite periodic strip and of an infinite periodic array of circular ABHs. In the latter case, collimation effects are observed. The transmissibility of finite ABH strips and arrays is also presented for comparison.

Convergences for robust bilevel polynomial programmes with applications

In this paper, we consider a bilevel polynomial optimization problem, where the constraint functions of both the upper-level and lower-level problems involve uncertain parameters. We employ the deterministic robust optimization approach to examine the bilevel polynomial optimization problem under data uncertainties by providing lower bound approximations and convergences of sum-of-squares (SOS) relaxations for the robust bilevel polynomial optimization problem. More precisely, we show that under the convexity of the lower-level problem and either the boundedness of the feasible set or the coercivity of the objective function, the global optimal values of SOS relaxation problems are lower bounds of the global optimal value of the robust bilevel polynomial problem and they converge to this global optimal value when the degrees of SOS polynomials in the relaxation problems tend to infinity. Moreover, an application to an electric vehicle charging scheduling problem with renewable energy sources demonstrates that using the proposed SOS relaxation schemes, we obtain more stable optimal values than applying a direct solution approach as the SOS relaxations are capable of solving these models involving data uncertainties in dynamic charging price and weather conditions.