Algebraic Form Research Articles

SAT Solving Using XOR-OR-AND Normal Forms

This paper introduces the XOR-OR-AND normal form (XNF) for logical formulas. It is a generalization of the well-known Conjunctive Normal Form (CNF) where literals are replaced by XORs of literals. As a first theoretic result, we show that every CNF formula is equisatisfiable to a formula in 2-XNF, i.e., a formula in XNF where each clause involves at most two XORs of literals. Subsequently, we present an algorithm which converts Boolean polynomials efficiently from their Algebraic Normal Form (ANF) to formulas in 2-XNF. Experiments with the cipher ASCON-128 show that cryptographic problems, which by design are based strongly on XOR-operations, can be represented using far fewer variables and clauses in 2-XNF than in CNF. In order to take advantage of this compact representation, new SAT solvers based on input formulas in 2-XNF need to be designed. By taking inspiration from graph-based 2-CNF SAT solving, we devise a new DPLL-based SAT solver for formulas in 2-XNF. Among others, we present advanced pre- and in-processing techniques. Finally, we give timings for random 2-XNF instances and instances related to key recovery attacks on round reduced ASCON-128, where our solver outperforms state-of-the-art alternative solving approaches.

Cluster consensus of finite-field networks with stochastic communication topology

Most of the multi-agent systems (MASs) in real life are with limited storage and communication capabilities; studies on such systems, including consensus problem, however have been mainly based on real number field models. In this article, the cluster consensus problem of MASs with stochastic communication topology over finite-field known as the cluster consensus of finite-field networks (FFNs), is considered. Using the semi-tensor product of matrices, the dynamics of FFNs with stochastic communication topology are equivalently converted into a logical algebraic form, which facilitates further studies. Then a permutation system is introduced. The relationship between cluster consensus of FFNs with stochastic communication topology and the set stability with probability one (SSPO) of the permutation system is revealed. In such an approach, some verifiable criteria can be derived for the cluster consensus of FFNs with stochastic communication topology, without requesting the connectivity of the whole network. Finally, an example is presented to validate the main results.

Braided Scalar Quantum Electrodynamics

AbstractWe formulate scalar electrodynamics in the braided ‐algebra formalism and study its perturbative expansion in the algebraic framework of Batalin–Vilkovisky quantization. We also confirm that UV/IR mixing is absent at one‐loop order in this noncommutative field theory, and that the non‐anomalous Ward‐Takahashi identities for the braided gauge symmetry are satisfied.

Finiteness of 3D higher spin gravity Landscape

Abstract We give Swampland constraints on the three dimensional Landscape of Anti-de Sitter higher spin gravity in the Chern-Simons formulation with connection valued in various split real forms of Lie algebras. We derive the finiteness conjecture by computing the upper bound on the rank of possible gauge groups then we refine it using the AdS distance conjecture. We discuss the implications of this Swampland constraint on the spectrum of higher spin gravity theories and we compare it with the gravitational exclusion principle required from BTZ black hole consideration to excerpt a constraint on the Chern-Simons level k.

Equivariant oriented homology of the affine Grassmannian

We generalize the property of small-torus equivariant K-homology of the affine Grassmannian to general oriented (co)homology theory in the sense of Levine and Morel. The main tool we use is the formal affine Demazure algebra associated to the affine root system. More precisely, we prove that the small-torus equivariant oriented cohomology of the affine Grassmannian satisfies the Goresky-Kottwitz-MacPherson (GKM) condition. We also show that its dual, the small-torus equivariant homology, is isomorphic to the centralizer of the equivariant oriented cohomology of a point in the formal affine Demazure algebra.

Study on the characteristics of fluid exciting force on impeller of high-speed centrifugal pump under the condition of poiseuille inflow

Abstract The paper adopted the turbulence model of Wall-Model Large Eddy Simulation with improved algebraic form (WMLES S-Omega) to carry out the study on the fluid exciting force on impeller of high-speed centrifugal pump under the conditions of Poiseuille inflow and uniform inflow. The external characteristics and the vibration displacement of the shaft of high-speed centrifugal pump are tested. The results indicated that it is the high matching that the test and numerical external data of high-speed centrifugal pump under the condition of Poiseuille inflow. The test results of vibration displacement of shaft match the fluid exciting force better in terms of waveform and phase difference under the condition of Poiseuille inflow. The contribution degree of force and torque from the shroud inner surface in the X and Y direction is 76.3%, 80.7%, 79.5% and 85% of the resultant force and torque, and its force from the inner and outer surfaces of shroud and hub in the Z direction is 94.3%, and its torque from blades’ surfaces in the Z direction is 98.3%. The study results can provide some theoretical support for the research of fluid exciting force and hydraulic optimization design of centrifugal pump.

Approximate Analytic Frequency of Strong Nonlinear Oscillator

In this paper, a new analytic expression for the frequency of vibration of a strong nonlinear polynomial-type oscillator is introduced. The method for frequency calculation is based on the transformation of the nonlinear oscillators into linear ones using the equality of their amplitudes and periods of vibration. The frequency of the linear oscillator is assumed to be the sum of frequencies corresponding to each nonlinearity in the original oscillator separately, i.e., the sum of frequencies of truly nonlinear oscillators. The obtained frequency is a complex function of amplitude, coefficient and order of nonlinearity. For simplification, the frequencies of the truly nonlinear oscillators are modified as power order functions of the exact frequency of the cubic oscillator which is linearly dependent on the amplitude of vibration. In this paper, the approximate frequency expression is developed for the harmonic any-order nonlinear oscillator and oscillators with the sum of polynomial nonlinearities. The accuracy of the obtained frequencies is tested on the examples of non-integer order nonlinear oscillators and also on a quadratic-cubic oscillator. The difference between the analytical and exact, numerically obtained results is negligible. The suggested approximate frequency expression has a simple algebraic form and is suitable for application by engineers and technicians.

ANALYSIS OF STUDENTS' ABILITY TO SOLVE MATHEMATICAL PROBLEMS ON CONTEXTUAL PROBLEMS INVOLVING ALGEBRAIC FORMS

This research is motivated by the fact that problem solving is a basic ability that every student must have. However, observations in the field show that students' mathematical problem-solving skills, especially in the material of algebraic form operations, are still relatively low. This study aims to describe students' mathematical problem-solving skills based on Polya's theory, which consists of four stages: (1) understanding the problem, (2) planning problem solving, (3) implementing the plan, and (4) re-examining the results. This study involved 26 ninth grade students as subjects. The method used was descriptive qualitative with data collection techniques including tests, interviews, and documentation, data analysis conducted based on four levels of problem-solving ability: excellent, good, sufficient, and deficient. The results showed that students' mathematical problem-solving ability on algebraic form operation material obtained as a whole was still relatively low with a scale of 42%. The implications of these findings indicate the need for the alication of more varied and contextualized teaching strategies to improve students' problem-solving skills. This research can be the basis for teachers in providing further assistance to students who are experiencing difficulties, focusing on improving skills in understanding and solving mathematical problems as a whole.

Application of Chelyshkov polynomials in solving stochastic model with fractional Brownian motion

Abstract This paper introduces a computational spectral collocation approach utilizing orthonormal Chelyshkov polynomials to estimate solutions for both linear and nonlinear stochastic differential equations containing fractional Brownian motion characterized by the Hurst parameter H. The method employs Newton–Cotes nodes to transform these integral equations into manageable linear and nonlinear algebraic forms. The reliability of the proposed method is established through theorems on error estimation and convergence analysis. Moreover, some examples are given to demonstrate the viability, credibility, coherence, and dependability of the approach. Also, a comparative evaluation is conducted to contrast the results obtained from the suggested approach with those produced by the spectral collocation method utilizing orthonormal Bernoulli polynomials. Furthermore, a 95 % confidence interval is constructed for the error mean, revealing that the approximations maintain high accuracy.

Development of Algebra Test Using the Item Response Theory Approach for Junior High School Students

This research aims to develop valid and reliable measuring tool for students' algebraic abilities that can be used in schools and the general public. The research follows a structured test development design, including stages such as preparing test specifications and items, field testing, revising items, and test development. The questions are aligned with the 2013 curriculum syllabus, ensuring relevance to educational standards. The test was given to 662 junior high school students in Kendari City, Indonesia, and their responses were analyzed using the item response theory (IRT) model with two logistic parameters: item difficulty level and item discriminatory power. The BILOG MG program was employed to estimate item and ability parameters. Before conducting item analysis with IRT, essential assumption tests were conducted, including unidimensional and model fit tests. The results of the development process, based on item analysis using BILOG MG, yielded 15 items covering various aspects of algebraic abilities. These items were derived from indicators such as recognizing algebraic forms, identifying elements within these forms, performing addition, subtraction, multiplication, and division operations on algebraic forms, presenting and solving real-world problems in algebraic contexts, and addressing contextual problems involving algebraic operations. The items demonstrated good fit with the model and exhibited an appropriate level of item difficulty and discriminatory power, making them suitable for use as a reliable assessment tool. Consequently, these developed tests are deemed effective for measuring students' foundational algebraic abilities.

A novel unified variational finite difference (VFD) solution method for optimal control problems

This article is aimed to propose a simple yet efficient unified numerical strategy for solving both linear and non-linear optimal control problems. To do so, the general form of quadratic performance index function and nonlinear state equations are considered first. Then, the idea of variational differential quadrature method is used to convert the integral/differential equations to the equivalent algebraic form. Since time is the only independent variable is this research, the finite difference method with an equally-spaced discretization scheme would be a more appropriate technique rather than the differential quadrature approach. So, the implemented numerical solution is called now as the variational finite difference method. The method of Lagrange multipliers is then utilized for minimization purpose and, as a result, the final set of nonlinear algebraic equations are obtained. Finally, the quadratic and triadic forms of non-linearity are considered and an explicit formulation is represented for the residual and Jacobian of the Newton-based iterative solution procedure. To demonstrate the accuracy and efficiency of the proposed approach in the quadratic optimal control area, several benchmark problems involving linear time-invariant, linear time-variant and nonlinear examples are successfully solved and the results are confirmed with those existed in the literature.

The Development of Powerpoint-Assisted Algebra Learning Media to Increase Students’ Mathematical Communication Ability on Junior High School

The matter of algebraic form is crucial to study because the matter of the algebraical form is not only related to other mathematical matters but also related to everyday life, but there are still many students who have difficulty in communicating the explanation of the origin of the algebraic form. This is because the learning media used by the teacher is not interactive, so it affects the student's understanding and communication skills. For this reason, the study aims to cultivate the interest in learning and cultivate students' mathematical communication skills by developing algebraic learning media supported by a PowerPoint application. The method used in this research is research and development, with the development of a modified 4-D model in three stages namely determining, designing, and developing. The tool used to measure the quality of the learning media is a validation questionnaire for measuring the validity and student response to strengthen practicality. The results of this study show that the quality of the learning media meets the valid criteria with an overall average score of 79% and From the test carried out against students obtained an average scoring of 76%. Thus, the development of algebra learning media supported by the PowerPoint application has good quality and can be used in mathematical learning.

Finite-time synchronization of proportional delay memristive competitive neural networks

The finite-time synchronization (FTS) is considered for proportional delay memristive competitive neural networks (PDMCNNs). By utilizing Lyapunov functional method and differential inclusion theory, two new criteria ensuring the FTS of PDMCNNs are established. These criteria with algebraic inequality forms are less complicated and easier to verify than the matrix inequality forms. In addition, the corresponding settling times have been estimated. Eventually, the effectiveness of the presented criteria and controllers is confirmed through two numerical examples, and one application about image encryption is provided.

The representation theory of Brauer categories II: Curried algebra

A representation of \mathfrak{gl}(V)=V \otimes V^{\ast} is a linear map \mu \colon \mathfrak{gl}(V) \otimes M \rightarrow M satisfying a certain identity. By currying, giving a linear map \mu is equivalent to giving a linear map a \colon V \otimes M \rightarrow V \otimes M , and one can translate the condition for \mu to be a representation into a condition on a . This alternate formulation does not use the dual of V and makes sense for any object V in a tensor category \mathcal{C} . We call such objects representations of the curried general linear algebra on V . The currying process can be carried out for many algebras built out of a vector space and its dual, and we examine several cases in detail. We show that many well-known combinatorial categories are equivalent to the curried forms of familiar Lie algebras in the tensor category of linear species; for example, the titular Brauer category is the curried form of the symplectic Lie algebra. This perspective puts these categories in a new light, has some technical applications, and suggests new directions to explore.

Predicting effects of tunnel throttling of annular freeway vehicular flow by a continuum model

Fluid flow throttling is common in industrial and building services engineering. Similar tunnel throttling of vehicular flow is caused by the abrupt number reduction of roadway lane, as the tunnel has a lower lane number than in the roadway normal segment. To predict the effects of tunnel throttling of annular freeway vehicular flow, a three-lane continuum model is developed. Lane III of the tunnel is completely blocked due to the need of tunnel rehabilitation, etc. There exists mandatory net lane-changing rate from lane III to lane II just upstream of the tunnel entrance, which is described by a model of random number generated through a golden section analysis. The net-changing rate between adjacent lanes is modeled using a lane-changing time expressed explicitly in algebraic form. This paper assumes that the annular freeway has a total length of 100 km, a two-lane tunnel of length 2 km with a speed limit of 80 km/h. The free flow speeds on lanes I, II and III are assumed to be 110, 100 and 90 km/h respectively. Based on the three-lane continuum model, numerical simulations of vehicular flows on the annular freeway with such a tunnel are conducted with a reliable numerical method of 3rd-order accuracy. Numerical results reveal that the vehicular flow has a smaller threshold of traffic jam formation in comparison with the case without tunnel throttling. Vehicle fuel consumption can be estimated by interpolation with time averaged grid traffic speed and an assumed curve of vehicle performance. The vehicle fuel consumption is lane number dependent, distributes with initial density concavely, ranging from 5.56 to 8.00 L. Tunnel throttling leads to an earlier traffic jam formation in comparison with the case without tunnel throttling.

Møller Maps for Dirac Fields in External Backgrounds

In this paper we study the foundations of the algebraic treatment of classical and quantum field theories for Dirac fermions under external backgrounds following the initial contributions already present in various places in the literature. The treatment is restricted to contractible spacetimes of globally hyperbolic nature in dimensions d≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 4$$\\end{document} and to external fields modelled with trivial principal bundles. In particular, we construct the classical Møller maps intertwining the configuration spaces of charged and uncharged fermions, and we show some of its properties in the case of a U(1) gauge charge. In the last part, as a first step towards a quantization of the theory, we explore the combination of the classical Møller maps with Hadamard bidistributions and prove that they are involutive isomorphisms (algebraically and topologically) between suitable (formal) algebras of functionals (observables) over the configuration spaces of charged and uncharged Dirac fields.

One-Bit Function Perturbation Impact on Robust Set Stability of Boolean Networks with Disturbances

This paper investigates the one-bit function perturbation (OBFP) impact on the robust set stability of Boolean networks with disturbances (DBNs). Firstly, the dynamics of these networks are converted into the algebraic forms utilizing the semi-tensor product (STP) method. Secondly, OBFP’s impact on the robust set stability of DBNs is divided into two situations. Then, by constructing a state set and defining an index vector, several necessary and sufficient conditions to guarantee that a DBN under OBFP can stay robust set stable unchanged are provided. Finally, a biological example is proposed to demonstrate the effectiveness of the obtained theoretical results.

A layer decomposition method for multi-layer elastic contact systems with interlayer Tresca friction

With the increasing demand for the accuracy of numerical simulation of pavement mechanics, the variational inequality model and its induced finite element method which can simulate the interlayer contact state becomes a potential solution. In this paper, a layer decomposition algorithm for solving variational inequality models of multi-layer elastic contact systems with interlayer Tresca friction conditions is studied. Continuous and discrete versions of the algorithm and their convergence theorems have been proposed and proved successively. Then, the algebraic form of the executable optimization algorithm and the numerical experimental results verify the practicability of the variational inequality model and its algorithm in the pavement mechanics modeling.

The specifics of the Galois field GF(257) and its use for digital signal processing

An algorithm of digital logarithm calculation for the Galois field GF(257)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$GF(257)$$\\end{document} is proposed. It is shown that this field is coupled with one of the most important existing standards that uses a digital representation of the signal through 256 levels. It is shown that for this case it is advisable to use the specifics of quasi-Mersenne prime numbers, representable in the form p=2n+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${p=2}^{n}+1$$\\end{document}, which includes the number 257. For fields GF(2n+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$GF({2}^{n}+1)$$\\end{document}, an alternating encoding can be used, in which non-zero elements of the field are displayed through binary characters corresponding to the numbers + 1 and − 1. In such an encoding, multiplying a field element by 2 is reduced to a quasi-cyclic permutation of binary symbols (the permuted symbol changes sign). Proposed approach makes it possible to significantly simplify the design of computing devices for calculation of digital logarithm and multiplication of numbers modulo 257. A concrete scheme of a device for digital logarithm calculation in this field is presented. It is also shown that this circuit can be equipped with a universal adder modulo an arbitrary number, which makes it possible to implement any operations in the field under consideration. It is shown that proposed digital algorithm can also be used to reduce 256-valued logic operations to algebraic form. It is shown that the proposed approach is of significant interest for the development of UAV on-board computers operating as part of a group.

EXCEPTIONAL SIMPLE REAL LIE ALGEBRAS AND VIA CONTACTIFICATIONS

Abstract In Cartan’s PhD thesis, there is a formula defining a certain rank 8 vector distribution in dimension 15, whose algebra of authomorphism is the split real form of the simple exceptional complex Lie algebra $\mathfrak {f}_4$ . Cartan’s formula is written in the standard Cartesian coordinates in $\mathbb {R}^{15}$ . In the present paper, we explain how to find analogous formulae for the flat models of any bracket generating distribution $\mathcal D$ whose symbol algebra $\mathfrak {n}({\mathcal D})$ is constant and 2-step graded, $\mathfrak {n}({\mathcal D})=\mathfrak {n}_{-2}\oplus \mathfrak {n}_{-1}$ . The formula is given in terms of a solution to a certain system of linear algebraic equations determined by two representations $(\rho ,\mathfrak {n}_{-1})$ and $(\tau ,\mathfrak {n}_{-2})$ of a Lie algebra $\mathfrak {n}_{00}$ contained in the $0$ th order Tanaka prolongation $\mathfrak {n}_0$ of $\mathfrak {n}({\mathcal D})$ . Numerous examples are provided, with particular emphasis put on the distributions with symmetries being real forms of simple exceptional Lie algebras $\mathfrak {f}_4$ and $\mathfrak {e}_6$ .