Basic Algebra Research Articles

The propagator of the inverted Caldirola-Kanai Oscillator

In this paper, we will derive the propagators for an inverted Caldirola-Kanai oscillator by the Feynman path integral, the Schwinger method, and the Dodonov method. In Feynman path inte-gral, the propagator can be calculated from the functional integrals while in the Schwinger method the propagator can be obtained from basic operator algebra and elementary integrations. In Dodonov method, the propagator can be derived from the application of the integrals of the motion of quantum systems. Furthermore, the connection between these methods is also discussed.

Prior knowledge of a calculus course: The impact of prior knowledge on students’ errors

We conducted a descriptive study among first-year engineering students at the Anton de Kom University of Suriname. We analyzed students’ errors regarding necessary prior knowledge in a calculus A exam. We found that the stage of the solution in which prior knowledge is required impacts the importance of prior knowledge. We also found that many errors concerned basic algebra and trigonometry concepts and skills. We concluded that even though the required prior knowledge concerns basic algebra and trigonometry, the stage of the solution in which prior knowledge is needed is of great importance.

Calculating toxic pressure for mixtures of endocrine disruptors

Incidence of autoimmune disorders, birth defects, and neurological diseases rose over the past 50 years due to increasing variety and quantity of pollutants. To date, there appear few methods capable to evaluate and predict mixture effects by endocrine disruptors (EDs). For the first time, we have developed calculus to determine mixture effects by all kinds of EDs.Our method uses the golden ratio ϕ and draws from bifurcation and chaos theory. Using also the concept of molecular mimicry, we developed the equation: effect=100%1+e5·∑KiCi−niϕ3. We successfully tested the equation using a range of cohort studies and biomarkers, and for different pollutants like heavy metals, thyroid hormone mimickants, chromate/chlorate, etc.The equation is simple enough to use with only minor prior knowledge and understanding of basic algebra. The method is universal and calculation is data 'light', requiring only pollutant concentrations [C], potencies K and an integer n for endocrinal involvement. This study offers a comprehensive framework to assess the health effects of pollutant exposure across diverse populations, envisioning far-reaching impact, and presenting practical examples and insights.

A Method for Specifying Complete Signature Randomization and an Algebraic Algorithm Based on It

To eliminate the limitations of signature randomization in known algebraic algorithms with a hidden group, the security of which is based on the computational complexity of solving large systems of power equations, a method for ensuring complete randomization is proposed. Based on this method, a new algorithm of the indicated type was developed, using a four-dimensional finite non-commutative associative algebra as an algebraic basis. We obtained estimates of the security of algorithms to direct attacks as well as from attacks based on known signatures, which confirm the effectiveness of the proposed signature randomization method. Due to the relatively small size and signature of the public and private keys, the developed algorithm is of interest as a potential practical post-quantum digital signature scheme.

Teachers’ mathematical work based on examples presented in the teaching of algebra in secondary education

Both international examination results and scientific evidence related to the teaching and learning of school algebra point to the necessity for analyzing mathematical practice in the classroom. In this context, we have formulated the research objective of characterizing the mathematical work that teachers favor based on the examples that they implement in the teaching of topics within the domain of secondary school algebra. To this end, this study is situated within the interpretive paradigm utilizing an instrumental collective case study approach. The study design entails the selection of two representative cases of mathematics teachers in Chilean schools, with the objective of understanding and describing their mathematical work in the secondary school classroom based on the examples they present. The results show that the first teacher utilizes varied examples that activate distinct forms of mathematical thinking, promoting the activation of both instrumental and discursive genesis. In contrast, the second teacher follows a more traditional approach, with the use of examples to illustrate and motivate, and a direct presentation of knowledge, activating primarily semiotic genesis, in which a transition between the numerical and the algebraic is lacking. Ultimately, the results highlight the importance of deeper consideration of the choice of examples in the teaching of algebra and how these examples can influence students’ learning. Furthermore, additional ideas are put forward for future work in this line of research associated with the use of examples in the classroom.

Mengeksplore aljabar awal di sekolah menengah: kombinasi analisis bibliometric dan meta analisis

The aims of this study is to analyze publications on early algebra in middle schools to contribute to the development of related literature. A total of 234 articles on early algebra published in various Scopus databases between 2013 and 2024 were retrieved and analyzed through a bibliometric analysis approach and 19 articles through a Comprehensive Meta Analysis (CMA). The results showed fluctuating trends in research related to early algebra in middle schools. The surge in publications using VOSviewer and RStudio from 2013 to 2024 did not align with a corresponding increase in average citations, contributing to a stagnation in the average citation per year. The noticeable upward trend in the volume of publications on early algebra since the beginning of 2013 indicates a dynamic and evolving research landscape in this field. This consistency suggests a widespread and sustained interest in early algebra research beyond the confines of a single country, further emphasizing its global relevance and significance. Integrating multimedia technology into early algebra instruction significantly enhances student learning outcomes. Moreover, access to technology and enhanced learning resources through technology integration can play a more crucial role in improving the effectiveness of early algebra learning in countries with lower income levels.

Eigenvalues of quantum Gelfand invariants

We consider the quantum Gelfand invariants which first appeared in a landmark paper by Reshetikhin et al. [Algebra Anal. 1(1), 178–206 (1989)]. We calculate the eigenvalues of the invariants acting in irreducible highest weight representations of the quantized enveloping algebra for gln. The calculation is based on Liouville-type formulas relating two families of central elements in the quantum affine algebras of type A.

Behind-the-horizon excitations from a single 2d CFT

In this work, we consider the atypical non-equilibrium state found in [1708.06328] which holographically represents a behind-the-horizon excitation in a black hole spacetime. The special feature of this state is that it looks like an equilibrium state when probed by a class of low-energy operators. First, we retrieve this property using the uniformization mapping in the limit of a large central charge, in the process we are able to derive rather than presume approximate thermal physics. Furthermore, in the large-c and high-energy limit, we realize these excitations as elements of the commutant algebra of a GNS-representation of the light operator algebra. Instead of analytically continuing a mixed heavy-light Euclidean correlator to a Lorentzian correlator, we identify the Euclidean correlator as a GNS-linear form and interpret the Lorentzian correlator as a vacuum expectation value of representatives of the light operator algebra on the GNS-vacuum.

Some Remarks Regarding Special Elements in Algebras Obtained by the Cayley–Dickson Process over Zp

In this paper, we provide some properties of k-potent elements in algebras obtained by the Cayley–Dickson process over Zp. Moreover, we find a structure of nonunitary ring over Fibonacci quaternions over Z3 and we present a method to encrypt plain texts, by using invertible elements in some of these algebras.

ChatGPT-generated help produces learning gains equivalent to human tutor-authored help on mathematics skills.

Authoring of help content within educational technologies is labor intensive, requiring many iterations of content creation, refining, and proofreading. In this paper, we conduct an efficacy evaluation of ChatGPT-generated help using a 3 x 4 study design (N = 274) to compare the learning gains of ChatGPT to human tutor-authored help across four mathematics problem subject areas. Participants are randomly assigned to one of three hint conditions (control, human tutor, or ChatGPT) paired with one of four randomly assigned subject areas (Elementary Algebra, Intermediate Algebra, College Algebra, or Statistics). We find that only the ChatGPT condition produces statistically significant learning gains compared to a no-help control, with no statistically significant differences in gains or time-on-task observed between learners receiving ChatGPT vs human tutor help. Notably, ChatGPT-generated help failed quality checks on 32% of problems. This was, however, reducible to nearly 0% for algebra problems and 13% for statistics problems after applying self-consistency, a "hallucination" mitigation technique for Large Language Models.

Opers on the projective line, Wronskian relations, and the Bethe Ansatz

It is well-known that the spectra of the Gaudin model may be described in terms of solutions of the Bethe Ansatz equations. A conceptual explanation for the appearance of the Bethe Ansatz equations is provided by appropriate G-opers: G-connections on the projective line with extra structure. In fact, solutions of the Bethe Ansatz equations are parameterized by an enhanced version of opers called Miura opers; here, the opers appearing have only regular singularities. Moreover, this geometric approach to the spectra of the Gaudin model provides a well-known example of the geometric Langlands correspondence. Feigin, Frenkel, Rybnikov, and Toledano Laredo have introduced an inhomogeneous version of the Gaudin model; this model incorporates an additional twist factor, which is an element of the Lie algebra of G. They exhibited the Bethe Ansatz equations for this model and gave an interpretation of the spectra in terms of opers with an irregular singularity. In this paper, we consider a new geometric approach to the study of the spectra of the inhomogeneous Gaudin model in terms of a further enhancement of opers called twisted Miura-Plücker opers. This approach involves a certain system of nonlinear differential equations called the qq-system, which were previously studied in [20] in the context of the Bethe Ansatz. We show that there is a close relationship between solutions of the inhomogeneous Bethe Ansatz equations and polynomial solutions of the qq-system and use this fact to construct a bijection between the set of solutions of the inhomogeneous Bethe Ansatz equations and the set of nondegenerate twisted Miura-Plücker opers. We further prove that as long as certain combinatorial conditions are satisfied, nondegenerate twisted Miura-Plücker opers are in fact Miura opers.

Wolfes model aka G2/I6-rational integrable model: g(2), g(3) hidden algebras and quartic polynomial algebra of integrals

One-dimensional 3-body Wolfes model with 2- and 3-body interactions also known as G2/I6-rational integrable model of the Hamiltonian reduction is exactly-solvable and superintegrable. Its Hamiltonian H and two integrals I1,I2, which can be written as algebraic differential operators in two variables (with polynomial coefficients) of the 2nd and 6th orders, respectively, are represented as non-linear combinations of g(2) or g(3) (hidden) algebra generators in a minimal manner. By using a specially designed MAPLE-18 code to deal with algebraic operators it is found that (H,I1,I2,I12≡[I1,I2]) are the four generating elements of the quartic polynomial algebra of integrals. This algebra is embedded into the universal enveloping algebra g(3). In turn, 3-body/A2-rational Calogero model is characterized by cubic polynomial algebra of integrals, it is mentioned briefly.

Lie Symmetries of the Wave Equation on the Sphere Using Geometry

A semilinear quadratic equation of the form Aij(x)uij=Bi(x,u)ui+F(x,u) defines a metric Aij; therefore, it is possible to relate the Lie point symmetries of the equation with the symmetries of this metric. The Lie symmetry conditions break into two sets: one set containing the Lie derivative of the metric wrt the Lie symmetry generator, and the other set containing the quantities Bi(x,u),F(x,u). From the first set, it follows that the generators of Lie point symmetries are elements of the conformal algebra of the metric Aij, while the second set serves as constraint equations, which select elements from the conformal algebra of Aij. Therefore, it is possible to determine the Lie point symmetries using a geometric approach based on the computation of the conformal Killing vectors of the metric Aij. In the present article, the nonlinear Poisson equation Δgu−f(u)=0 is studied. The metric defined by this equation is 1 + 2 decomposable along the gradient Killing vector ∂t. It is a conformally flat metric, which admits 10 conformal Killing vectors. We determine the conformal Killing vectors of this metric using a general geometric method, which computes the conformal Killing vectors of a general 1+(n−1) decomposable metric in a systematic way. It is found that the nonlinear Poisson equation Δgu−f(u)=0 admits Lie point symmetries only when f(u)=ku, and in this case, only the Killing vectors are admitted. It is shown that the Noether point symmetries coincide with the Lie point symmetries. This approach/method can be used to study the Lie point symmetries of more complex equations and with more degrees of freedom.

Primitive elements of free non-associative algebras over finite fields

The representation of elements of free non-associative algebras as a set of multidimensional tables of coefficients is defined. An operation for finding partial derivatives for elements of free non-associative algebras in the same form is considered. Using this representation, a criterion of primitivity for elements of lengths 2 and 3 in terms of matrix ranks, as well as a primitivity test for elements of arbitrary length, is derived. This test makes it possible to estimate the number of primitive elements in free non-associative algebras with two generators over a finite field. The proposed representation allows us to optimize algorithms for symbolic computations with primitive elements. Using these algorithms, we find the number of primitive elements of length 4 in a free non-associative algebra of rank 2 over a finite field.

Hilbert’s Nullstellensatz

Hilbert’s Nullstellensatz proved by him in 1890 is one of the basic results in modern algebraic geometry. We give various statements and proofs of this theorem all of which are used in algebraic geometry. All notions and facts outside the basic algebra course are explained in the paper.

Automorphism groups of affine varieties consisting of algebraic elements

Given an affine algebraic variety X X , we prove that if the neutral component A u t ∘ ( X ) \mathrm {Aut}^\circ (X) of the automorphism group consists of algebraic elements, then it is nested, i.e., is a direct limit of algebraic subgroups. This improves our earlier result (see Perepechko and Regeta [Transform. Groups 28 (2023), pp. 401–412]). To prove it, we obtain the following fact. If a connected ind-group G G contains a closed connected nested ind-subgroup H ⊂ G H\subset G , and for any g ∈ G g\in G some positive power of g g belongs to H H , then G = H G=H .

On the combinatorial rank of quantum groups of type F4

In this paper, we calculate the combinatorial rank of the positive part [Formula: see text] of the multiparameter version of the small Lusztig quantum group, where [Formula: see text] is a simple Lie algebra of type [Formula: see text]. For this purpose, we also present the basis and relations of the quantum algebra, calculate the coproduct of its PBW-generators and determine all skew-primitive elements of this quantum group. In conclusion, we obtain that the combinatorial rank of [Formula: see text] is 4.

Contributions of the anthropological theory of the didactic to the epistemological programme of research in mathematics education

In the early 1980s, the theory of didactic situations put the analysis of mathematical activity at the core of the didactic analysis, thus initiating a new research programme in mathematics education: the epistemological programme. This paper describes and interprets some of the contributions of the anthropological theory of the didactic (ATD) to the development of this programme. These contributions come from the new institutional and ecological perspective provided by the ATD to approach the didactic problem and from the significance it assigns to macrodidactic phenomena characterised by didactic transposition. They are illustrated with research works involving different fields of school mathematics in which the ATD team led by Spanish researchers played a central role: research in school algebra and its relations with functional modelling, differential calculus, and real numbers. The paper concludes by briefly describing the contributions of the ATD to the dialogue between didactic theories, another domain that has notable Spanish contributions.

Primitive Elements of Free Non-associative Algebras over Finite Fields

Primitive Elements of Free Non-associative Algebras over Finite Fields

Fast and exact fixed-radius neighbor search based on sorting.

Fixed-radius near neighbor search is a fundamental data operation that retrieves all data points within a user-specified distance to a query point. There are efficient algorithms that can provide fast approximate query responses, but they often have a very compute-intensive indexing phase and require careful parameter tuning. Therefore, exact brute force and tree-based search methods are still widely used. Here we propose a new fixed-radius near neighbor search method, called SNN, that significantly improves over brute force and tree-based methods in terms of index and query time, provably returns exact results, and requires no parameter tuning. SNN exploits a sorting of the data points by their first principal component to prune the query search space. Further speedup is gained from an efficient implementation using high-level basic linear algebra subprograms (BLAS). We provide theoretical analysis of our method and demonstrate its practical performance when used stand-alone and when applied within the DBSCAN clustering algorithm.