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This research addresses the issue of how to support students' representational fluency—the ability to create, move within, translate across, and derive meaning from external representations of mathematical ideas. The context of solving linear equations in a combined computer algebra system (CAS) and paper‐and‐pencil classroom environment is targeted as a rich and pressing context to study this issue. We report results of a collaborative teaching experiment in which we designed for and tested a functions approach to solving equations with ninth‐grade algebra students, and link to results of semi‐structured interviews with students before and after the experiment. Results of analyzing the five‐week experiment include instructional supports for students' representational fluency in solving linear equations: (a) sequencing the use of graphs, tables, and CAS feedback prior to formal symbolic transpositions, (b) connecting solutions to equations across representations, and (c) encouraging understanding of equations as equivalence relations that are sometimes, always, or never true. While some students' change in sophistication of representational fluency helps substantiate the productive nature of these supports, other students' persistent struggles raise questions of how to address the diverse needs of learners in complex learning environments involving multiple tool‐based representations. [ABSTRACT FROM AUTHOR]

We derive a new representation for the exact convergence factor of classical two-level and two-grid preconditioners. Based on this result, we establish necessary and sufficient conditions for constructing the components of efficient algebraic multigrid (AMG) methods. The relation of the sharp estimate to the classical two-level hierarchical basis methods is discussed as well. Lastly, as an application, we give an optimal two-grid convergence proof of a purely algebraic ‘window’-AMG method. Published in 2005 by John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The author briefly reviews studies on the math teaching of secondary school students with mild intellectual disabilities. Then, the author demonstrates ways to teach secondary‐level mathematics to students with mild intellectual disabilities. In this article, readers will learn about how to use manipulatives, diagrams, and gestures to support students' thinking. Readers will also learn how to make connections between new and challenging math content to students' experiences inside and outside of school to support them as they think through mathematics. [ABSTRACT FROM AUTHOR]

Peter Neumann was born in Oxford on 28 December 1940, the first son of mathematicians Bernhard and Hanna Neumann, who came to the UK from Germany in the 1930s. Peter obtained his DPhil in 1963 under the supervision of Graham Higman, after which he was appointed to a fellowship at Queen's and a university lectureship. He spent his entire career at Oxford, retiring in 2008. Peter's lifelong contribution to mathematics in the UK and worldwide was monumental and wide‐ranging: through his research in algebra and the history of algebra; his supervision of over 40 doctoral students, many of whom went on to have distinguished academic careers; his extensive service to the London Mathematical Society; and his enormous contribution to mathematics education. Peter was a leading figure in algebra for over 50 years, publishing around 100 papers and books on varieties of groups, soluble groups, group enumeration, permutation groups, computational algebra, and the history of algebra. He was a great collaborator, publishing with 38 different co‐authors, and holding visiting positions at many places around the world. His contributions to research and scholarship were recognised by the London Mathematical Society with the award of the Senior Whitehead Prize in 2003 and with the joint LMS–IMA David Crighton Medal in 2012. Peter also made an enormous contribution to mathematics education in the UK. He was the founding Chairman of UK Mathematics Trust, serving from 1996 to 2005. During his period as chairman, Peter led UKMT in taking on the staging of the 2002 International Mathematical Olympiad. For his services to mathematics education, Peter was awarded an OBE in 2008. Peter was a superb lecturer, writer and expositor, and delightful company with a sparkling sense of humour. Above all, Peter is remembered for his kindness and generosity, his infectious and inspirational enthusiasm, and his boundless energy. [ABSTRACT FROM AUTHOR]

Let $p$ be a prime for which the congruence group $\Gamma_0(p)^*$ is of genus zero, and let $j_p^*$ be the corresponding Hauptmodul. Let $f$ be a nearly holomorphic modular form of weight 1/2 on $\Gamma_0(4p)$ which satisfies some congruence condition on its Fourier coefficients. We interpret $f$ as a vector valued modular form. Applying Borcherds lifting of vector valued modular forms we construct infinite products associated to $j_p^*$ and relate them to Zagier's trace formula for the singular values of $j_p^*$. [ABSTRACT FROM AUTHOR]

This document is an editorial summarizing the Applied Geometric Algebras in Computer Science and Engineering (AGACSE) conference held in Brno, Czech Republic in September 2021. The conference aimed to promote the use of geometric algebra in fields such as image processing, robotics, and quantum computing. The conference proceedings were published in the journal Mathematical Methods in the Applied Sciences. The editorial provides a list of accepted papers, covering topics such as applied geometry, technological applications, algebra, and quantum phenomena. One specific paper explores the use of geometric algebra in teaching rotations through neural networks. The document is a compilation of research papers showcasing the applications of geometric algebra in various fields, including robotics, control systems, image processing, cryptography, and physics. Each paper presents a specific problem or application and proposes a unique approach or solution using geometric algebra. The authors compare their methods with existing techniques and provide mathematical analysis to support their claims. Overall, the papers demonstrate the versatility and effectiveness of geometric algebra in different domains. [Extracted from the article]

Five widely used metrics of bioaccumulation in fish are defined and discussed, namely the octanol-water partition coefficient ( KOW), bioconcentration factor (BCF), bioaccumulation factor (BAF), biomagnification factor (BMF), and trophic magnification factor (TMF). Algebraic relationships between these metrics are developed and discussed using conventional expressions for chemical uptake from water and food and first-order losses by respiration, egestion, biotransformation, and growth dilution. Two BCFs may be defined, namely as an equilibrium partition coefficient KFW or as a nonequilibrium BCFK in which egestion losses are included. Bioaccumulation factors are shown to be the product of the BCFK and a novel equilibrium multiplier M containing 2 ratios, namely, the diet-to-water concentration ratio and the ratio of uptake rate constants for respiration and dietary uptake. Biomagnification factors are shown to be proportional to the lipid-normalized ratio of the predator/prey values of BCFK and the ratio of the equilibrium multipliers. Relationships with TMFs are also discussed. The effects of chemical hydrophobicity, biotransformation, and growth are evaluated by applying the relationships to a range of illustrative chemicals of varying KOW in a linear 4-trophic-level food web with typical values for uptake and loss rate constants. The roles of respiratory and dietary intakes are demonstrated, and even slow rates of biotransformation and growth can significantly affect bioaccumulation. The BCFKs and the values of M can be regarded as the fundamental determinants of bioaccumulation and biomagnification in aquatic food webs. Analyzing data from food webs can be enhanced by plotting logarithmic lipid-normalized concentrations or fugacities as a linear function of trophic level to deduce TMFs. Implications for determining bioaccumulation by laboratory tests for regulatory purposes are discussed. Environ Toxicol Chem 2013;32:1459-1466. © 2013 SETAC [ABSTRACT FROM AUTHOR]

A Picard–Borel algebra is a commutative unital complex algebra A$A$ such that every family of pairwise linearly independent invertible elements of A$A$ is linearly independent, and a Picard–Borel ideal I$I$ in a commutative complex unital algebra A$A$ is an ideal of I$I$ of A$A$ such that the quotient algebra A/I$A/I$ is a Picard–Borel algebra. In a preliminary paper, the author proved that every commutative unital Fréchet algebra which is a Picard–Borel algebra is an integral domain. The main result of the present paper is that all Picard–Borel ideals in commutative unital Fréchet algebras are prime. This result seems to be relevant for Michael's problem, since dense Picard–Borel ideals could play a role in the construction of discontinuous characters on some commutative unital Fréchet algebras. [ABSTRACT FROM AUTHOR]

The paper written many years ago by Resco and Small, constructed an example in prime characteristic of a finitely generated Noetherian K$K$‐algebra that does not necessarily remain Noetherian under field extensions. After this work was completed, Yuri Medvedev, then at the University of Ottawa, visited Resco at the University of Oklahoma, read the paper, and suggested an approach using Jordan algebras that would yield a similar example in characteristic 0. This was announced in the paper by Resco and Small, but to our knowledge the details were never published nor even entirely verified. We take the opportunity to do this now, but stress that the key sequence {an∣n=1,2,3,...}$\lbrace a_n\mid n=1,2,3,\ldots \rbrace$ and its relevant properties are due to Medvedev. Also, Resco suggested avoiding Jordan algebras and working directly and ring theoretically with the exterior algebra. This is the approach we take. [ABSTRACT FROM AUTHOR]

We prove that the localization Cw of the monoidal category Cw is rigid, and the category Cw,v admits a localization via a real commuting family of central objects. For a quiver Hecke algebra R and an element w in the Weyl group, the subcategory Cw of the category R-gmod of finite-dimensional graded R-modules categorifies the quantum unipotent coordinate ring Aq(n(w)). In the previous paper, we constructed a monoidal category Cw such that it contains Cw and the objects {(M(wΛi,Λi) i ∈ 1} corresponding to the frozen variables are invertible. In this paper, we show that there is a monoidal equivalence between the category Cw and (Cw-1)rev. Together with the already known left-rigidity of Cw, it follows that the monoidal category Cw is rigid. If v ≤ w in the Bruhat order, there is a subcategory Cw,v of Cw of that categorifies the doubly-invariant algebra N'(w)ℂ[N]N(v). We prove that the family [M(wΛi,vΛi))i∈I of simple R-module forms a real commuting family of graded central objects in the category Cw,v so that there is a localization Cw,v of Cw,v in which M(wΛi,vΛi) are invertible. Since the localization of the algebra N'(w)ℂ[N]N(v) by the family of the isomorphism classes of M(wΛi,vΛi) is isomorphic to the coordinate ring ℂ[Rw,v] of the open Richardson variety associated with w and v, the localization Cw,v categorifies the coordinate ring ℂ[Rw,v]. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a generalization of Vieta theorem (Vieta's formulas) to the case of Clifford geometric algebras. We compare the generalized Vieta formulas with the ordinary Vieta formulas for characteristic polynomial containing eigenvalues. We discuss Gelfand–Retakh noncommutative Vieta theorem and use it for the case of geometric algebras of small dimensions. We introduce the notion of a simple basis‐free formula for a determinant in geometric algebra and prove that a formula of this type exists in the case of arbitrary dimension. Using this notion, we present and prove generalized Vieta theorem in geometric algebra of arbitrary dimension. The results can be used in symbolic computation and various applications of geometric algebras in computer science, computer graphics, computer vision, physics, and engineering. [ABSTRACT FROM AUTHOR]

In this paper, we develop the Hermitian refinement of symplectic Clifford analysis, by introducing a complex structure 핁 on the canonical symplectic manifold (ℝ2n,ω0)$$ \left({\mathrm{\mathbb{R}}}^{2n},{\omega}_0\right) $$. This gives rise to two symplectic Dirac operators Ds$$ {D}_s $$ and Dt$$ {D}_t $$ (in the sense of Habermann), leading to a u(n)$$ \mathfrak{u}(n) $$‐invariant system of equations on ℝ2n$$ {\mathrm{\mathbb{R}}}^{2n} $$. We discuss the solution space for this system, culminating in a Fischer decomposition for the space of (harmonic) polynomials on ℝ2n$$ {\mathrm{\mathbb{R}}}^{2n} $$ with values in the symplectic spinors. To make this decomposition explicit, we will construct the associated embedding factors using a transvector algebra. [ABSTRACT FROM AUTHOR]

For any finite group G, the Terwilliger algebra T(G) of the group association scheme satisfies the following inclusions: T0(G)⊆T(G)⊆T˜(G), where T0(G) is a specific vector space and T˜(G) is the centralizer algebra of the permutation representation of G induced by the action of conjugation. The group G is said to be triply transitive if T0(G)=T˜(G). In this paper, we determine the dimensions of T0(G) and T˜(G) for G being Tn,k=〈a,b∣a2n=1,an=b2,bab−1=ak〉, Cn⋊Cp and Cp⋊Cn, and show that Tn,k,Cn⋊C2 and C3⋊C2n are triply transitive. Additionally, we give a complete characterization of the Wedderburn components of the Terwilliger algebras of Tn,k, Cn⋊Cp and Cp⋊Cn when they are triply transitive. [ABSTRACT FROM AUTHOR]

Important correspondences in representation theory can be regarded as restrictions of the Morita–Tachikawa correspondence. Moreover, this correspondence motivates the study of many classes of algebras like Morita algebras and gendo‐symmetric algebras. Explicitly, the Morita–Tachikawa correspondence describes that endomorphism algebras of generators–cogenerators over finite‐dimensional algebras are exactly the finite‐dimensional algebras with dominant dimension at least two. In this paper, we introduce the concepts of quasi‐generators and quasi‐cogenerators that generalise generators and cogenerators, respectively. Using these new concepts, we present higher versions of the Morita–Tachikawa correspondence that take into account relative dominant dimension with respect to a self‐orthogonal module with arbitrary projective and injective dimensions. These new versions also hold over Noetherian algebras that are finitely generated and projective over a commutative Noetherian ring. [ABSTRACT FROM AUTHOR]

Let F$F$ be a field of characteristic zero and G$G$ a finite abelian group. In this paper, we prove that an affine variety of G$G$‐graded PI‐algebras is minimal if and only if it is generated by a graded algebra UT(A1,⋯,Am;γ)$UT(A_1,\dots,A_m;\gamma)$ of upper block triangular matrices where A1,⋯,Am$A_1,\dots,A_m$ are finite‐dimensional G$G$‐simple algebras. [ABSTRACT FROM AUTHOR]

In this exploratory case study, the researchers conducted a descriptive, qualitative microanalysis of the tutoring of two eighth grade students with learning disabilities while solving algebra problems. The researchers analyzed the participants' problem solving tendencies and interventions that helped the students succeed. One of the students, Nicole, demonstrated considerable capability at supporting her memory and processing by carefully and neatly writing out her steps for solving equations on paper. The other participant, Rachel, struggled at times to organize her thinking processes on paper and solve equations due to poor handwriting legibility, math anxiety and issues of confidence as a learner of mathematics that all seemed to be inter‐related. Both students demonstrated success with solving algebra equations, but Rachel required intensive intervention for addressing her difficulties with math anxiety and visual‐motor integration, while Nicole mostly required mathematics instruction designed to capitalise on her success at creating useful visual representations of her thought processes. [ABSTRACT FROM AUTHOR]

This paper presents an algorithm of computing the robust viability kernel for discrete‐time switched systems with arbitrary switching rule and bounded disturbance based on the robust one‐step set and Pontryagin difference. The algorithm can be implemented by performing the relevant set computations using polyhedral algebra and computational geometry software when switched systems are linear. In addition, we propose a method of computing inner approximation of the robust viability kernel for switched systems based on the set theory. The convergence of the iterative algorithm is proved using a null controllable set. Finally, two examples are provided to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper we show that for a vector space Vd$V_d$ of dimension d$d$ there exists a linear map detS2:Vd⊗d(2d−1)→k$det^{S^2}:V_d^{\otimes d(2d-1)}\rightarrow k$ with the property that detS2(⊗1⩽i

Summary: In this paper, we introduce a method for reasoning with words based on hedge algebra using linguistic cognitive map (핃ℂ필). Our model consists of a set of vertices and edges whose values are linguistic variables that are constrained by a linguistic lattice. The algorithm for the system studied consists of fuzzification, reasoning, and defuzzification. Further, the total effect between any two vertices in the 핃ℂ필 is computed. [ABSTRACT FROM AUTHOR]

This paper presents a bicoprime factorization (BCF) approach to deal with various stability problems for linear discrete time‐varying (LTV) systems within the framework of nest algebra. Based on the bicoprime factorization, we give necessary and sufficient conditions for (strongly) simultaneous stabilization of three systems, which generalize similar results for systems by coprime factorization (CF). Finally, when the bicoprime factorization for the plant and controller has additive perturbations, we derive sufficient conditions using the norm inequality for robust stability of the system. [ABSTRACT FROM AUTHOR]

Recently, adaptive filtering algorithms have attracted much more attention in the field of signal processing. By studying the shortcoming of the traditional real‐valued fixed step size adaptive filtering algorithm, this paper proposed the novel approach to adaptive filtering with variable step size based on Sigmoid function and geometric algebra (GA). First, the proposed approach to adaptive filtering with variable step size based on geometric algebra represents the multi‐dimensional signal as a GA multi‐vector for the vectorization process. Second, the proposed approach to adaptive filtering with variable step size based on geometric algebra solves the contradiction between the steady‐state error and the convergence rate by establishing a non‐linear function relationship between the step size and the error signal. Finally, the experimental results demonstrate that the proposed approach to adaptive filtering with variable step size based on geometric algebra achieves better performance than that of the existing adaptive filtering algorithms. [ABSTRACT FROM AUTHOR]

Several of the most fundamental transformations widely used by the power engineering community are strongly based on geometrical considerations. Clifford's geometric algebra (GA) is the natural language for describing concepts in Euclidean geometry. In this work, we show how Clarke, Park, and Depenbrock's FBD transformations can be derived by imposing orthogonality on the voltage and current vectors defined in a Euclidean space by using GA. This paper presents these transformations as spatial‐like rotations and projections by means of the use of special algebraic objects named GA rotors. We prove that there is no need to use complex numbers nor matrices to perform the mentioned transformations and to provide geometrical intuition for electrical quantities and their transformations. Furthermore, power properties can be described using GA terminology by means of the proposed geometric power. The manipulation of the geometric power allows a useful current decomposition for a variety of applications such active filtering, frequency estimation, or electrical machinery. We provide in this paper an alternative approach focused on power systems under the paradigm of GA. [ABSTRACT FROM AUTHOR]

The G8,2 geometric algebra, also called the double conformal/Darboux cyclide geometric algebra (DCGA), has entities that represent conic sections. DCGA also has entities that represent planar sections of Darboux cyclides, which are called cyclidic sections in this paper. This paper presents these entities and many operations on them, in terms of algebraic expressions, computer code, and many examples of computer‐generated graphics. Operations include reflection, projection, rejection, and intersection with respect to spheres and planes. Other operations include rotation, translation, and dilation. Possible applications are introduced, which include orthographic and perspective projections of conic sections onto view planes, which may be of interest in computer graphics or other computational geometry subjects [ABSTRACT FROM AUTHOR]

In this paper, we prove that the hyperfocal subalgebra of a block with an abelian defect group and a cyclic hyperfocal subgroup is Rickard equivalent to the group algebra of the semidirect of the hyperfocal subgroup by the inertial quotient of the block. In particular, the hyperfocal subalgebra is a Brauer tree algebra, which is analogous to the structure of blocks with cyclic defect groups. As a consequence, we show that Broué's abelian defect group conjecture holds for blocks with cyclic hyperfocal subgroups. [ABSTRACT FROM AUTHOR]

The automorphism group of a quantised coordinate algebra is usually much smaller than that of its classical counterpart. Nevertheless, these automorphism groups are often very difficult to calculate. In this paper, we calculate the automorphism group of the quantum grassmannian in the case that the deformation parameter is not a root of unity. The main tool employed is the dehomogenisation equality which shows that a localisation of the quantum grassmannian is equal to a skew Laurent extension of quantum matrices. This equality is used to connect the automorphism group of the quantum grassmannian with that of quantum matrices, where the automorphism group is known. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the two‐dimensional short‐time quaternion offset linear canonical transform (ST‐QOLCT), which is a generalization of the classical short‐time offset linear canonical transform (ST‐OLCT) in quaternion algebra setting. Several useful properties of the ST‐QOLCT are obtained from the properties of the ST‐QOLCT kernel. Based on the properties of the ST‐QOLCT and the convolution and correlation operators associated with QOLCT, we derive convolution and correlation theorems for the ST‐QOLCT. Finally, some potential applications of the ST‐QOLCT are introduced. [ABSTRACT FROM AUTHOR]

Additive light field displays are transparent autostereoscopic three‐dimensional displays without backlights, thus suitable for augmented reality applications. However, when the parallax between viewpoint images becomes large with the increase of viewing angle, the optimization algorithm is hard to handle too many dissimilarities evenly on all viewpoints, resulting in poor reconstructed quality. This paper presents an additive light field display using the weighted simultaneous algebraic reconstruction technique with viewing angle‐dependent weight distribution functions. We constrain the optimization to deliver a reconstructed light field of high image quality for viewpoints of large weight. When the proposed method is applied, with a wide dynamic viewing angle of 57° × 43°, the tracked views' peak signal‐to‐noise ratio exceeds 30 dB with only two additive display layers. [ABSTRACT FROM AUTHOR]

Summary: This paper presents and discusses the usage of Geometric Algebra (GA) for the analysis of electrical alternating current (AC) circuits. The potential benefits of this novel approach are highlighted in the study of linear and nonlinear circuits with sinusoidal and non‐sinusoidal sources in the frequency domain, which are important issues in electrical engineering undergraduate courses. The analysis and understanding of how AC circuits operate in steady state are of a paramount importance for all the electrical engineers and practitioners around the world. Typically, lecturers of most undergraduate courses teach circuit theory using complex phasors, vector calculus, or linear algebra. However, these approaches have some important limitations in practice, which requires the development of strategies to improve teaching‐learning process related to AC circuit analysis. By formulating a new mathematical framework, the paper presents and discusses how GA can be of help in this approach. It is also described how the results obtained by using GA are validated using computer‐based simulations. It is highlighted how the proposed teaching methodology based on GA theory can be effective in helping students learn AC circuit analysis since it has several potential benefits derived from its simplicity, compactness, and use of easily identifiable geometrical concepts. [ABSTRACT FROM AUTHOR]

As is well‐known, on an Arens regular Banach algebra all continuous functionals are weakly almost periodic. In this paper, we show that ℓ1‐bases which approximate upper and lower triangles of products of elements in the algebra produce large sets of functionals that are not weakly almost periodic. This leads to criteria for extreme non‐Arens regularity of Banach algebras in the sense of Granirer. We find in particular that bounded approximate identities (bai's) and bounded nets converging to invariance (TI‐nets) both fall into this approach, suggesting that this is indeed the main tool behind most known constructions of non‐Arens regular algebras. These criteria can be applied to the main algebras in harmonic analysis such as the group algebra, the measure algebra, the semigroup algebra (with certain weights) and the Fourier algebra. In this paper, we apply our criteria to the Lebesgue‐Fourier algebra, the 1‐Segal Fourier algebra and the Figà‐Talamanca Herz algebra. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamic response of an elastic single-degree-of-freedom oscillator coupled with a rocking wall. Both configurations of a stepping rocking wall and a pinned rocking wall that have been reported in the literature are examined. The full nonlinear equations of motions are derived, and the paper shows through a comprehensive parametric analysis that the coupling with a rocking wall has mixed results on suppressing the dynamic response of the elastic oscillator. The stepping rocking wall is most effective in suppressing displacements of relative flexible structures with a heavier wall being most effective. In contrast, the pinned wall amplifies the displacements along a wide range of the spectrum with a heavier wall being most detrimental. This happens partly because in a pinned wall the moment from its weight works against stability. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Summary: This paper proposes an algebraic system, verification algebra (VA), for reducing the number of component combinations to be verified in multi‐tenant architecture (MTA). MTA is a design architecture used in SaaS (Software‐as‐a‐Service) where a tenant can customize its applications by integrating services already stored in the SaaS databases or newly supplied services. Similar to SaaS, VaaS (Verification‐as‐a‐Service) is a verification service in a cloud that leverages the computing power offered by a cloud environment with automated provisioning, scalability and service composition. In VaaS architecture, however, there is a challenging problem called 'combinatorial explosion' that it is difficult to verify a large number of compositions constructed by both quantities of components and various combination structures even with computing resources in cloud. This paper proposes rules to emerge combinations status for future verification, on the basis of the existing results. Both composition patterns and properties are considered and analysed in VA rules. [ABSTRACT FROM AUTHOR]

This paper constructs homogeneous affine sandwich cellular bases of weighted KLRW algebras in types BZ⩾0$B_{\mathbb {Z}_{\geqslant 0}}$, A2·e(2)$A^{(2)}_{2\cdot e}$, De+1(2)$D^{(2)}_{e+1}$. Our construction immediately gives homogeneous sandwich cellular bases for the finite‐dimensional quotients of these algebras. Since weighted KLRW algebras generalize KLR algebras, we also obtain bases and cellularity results for the (infinite and finite‐dimensional) KLR algebras. [ABSTRACT FROM AUTHOR]

The flipped classroom (FC) is becoming an increasingly popular teaching method in mathematics education. However, few studies have rigorously evaluated its effectiveness, and less so in countries where students' level of mathematics achievement is low. In this study, we evaluated the impact of an FC intervention in Saudi Arabia, the country with the lowest level of maths achievement in the last iteration of the Trends International Mathematics and Science Study TIMSS. A total of 281 secondary school students received eight weeks of algebra training using either FC or traditional instruction, and were tested on algebra problems taken from past national standardised tests, as well as on a measure of self‐efficacy. Students who received the intervention showed higher self‐efficacy but no significant difference in maths achievement was observed. Students' and teachers' perceptions of the intervention were positive. Practitioner notes: What is already known about this topic The flipped classroom is an increasingly popular teaching method in mathematics education.There is evidence that the method is beneficial for students.However, there have been very few rigorous evaluation studies of flipped classrooms and most were conducted in the US, Taiwan and China.Whether the beneficial impact of flipped classrooms can be generalised to other countries, particularly countries with low proficiency in mathematics, remains unclear. What this paper adds We evaluated the impact of the flipped classroom in Saudi Arabia, a country with low proficiency in mathematics and limited technological resources.We used a rigorous design (with a control group) and educationally relevant outcome measures.We found that a flipped classroom can have a positive impact on students' self‐efficacy, and that the method was perceived positively by both teachers and students.Based on interview data, we also documented the perceptions and concerns of the participating teachers. Implications for practice and/or policy Flipped classrooms showed promising results in a context which differs markedly from previous evaluations, suggesting that the method can be beneficial in a range of contexts.This positive impact was observed despite the intervention being relatively short (six weeks) and implemented by teachers previously unfamiliar with the flipped classroom method. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of multigraded algebras and multigraded linear series. For an Ns$\mathbb {N}^s$‐graded algebra A$A$, we define and study its volume function FA:N+s→R$F_A:\mathbb {N}_+^s\rightarrow \mathbb {R}$, which computes the asymptotics of the Hilbert function of A$A$. We relate the volume function FA$F_A$ to the volume of the fibers of the global Newton–Okounkov body Δ(A)$\Delta (A)$ of A$A$. Unlike the classical case of standard multigraded algebras, the volume function FA$F_A$ is not a polynomial in general. However, in the case when the algebra A$A$ has a decomposable grading, we show that the volume function FA$F_A$ is a polynomial with nonnegative coefficients. We then define mixed multiplicities in this case and provide a full characterization for their positivity. Furthermore, we apply our results on multigraded algebras to multigraded linear series. Our work recovers and unifies recent developments on mixed multiplicities. In particular, we recover results on the existence of mixed multiplicities for (not necessarily Noetherian) graded families of ideals and on the positivity of the multidegrees of multiprojective varieties. [ABSTRACT FROM AUTHOR]

In this paper, we address a question concerning nilpotent Frobenius actions on Rees algebras and associated graded rings. We prove a nilpotent analog of a theorem of Huneke for Cohen–Macaulay singularities. This is achieved by introducing a depth‐like invariant which captures as special cases Lyubeznik's F‐depth and the generalized F‐depth from Maddox–Miller and is related to the generalized depth with respect to an ideal. We also describe several properties of this new invariant and identify a class of regular elements for which weak F‐nilpotence deforms. [ABSTRACT FROM AUTHOR]

In this work, we propose a geometric algebra‐based variation of a well‐known threshold secret‐sharing scheme introduced by Adi Shamir in 1979. Secret sharing is a cryptographic primitive which allows a secret input to be divided into multiple shares which are then sent to a collection of parties. The shares are generated so that only "authorized" sets of shares can reconstruct the secret. In Shamir's scheme, any sufficiently large set of shares can reconstruct the secret. The minimum number of shares which can obtain the secret is called the threshold, and any number of shares smaller than the threshold reveals nothing about the secret. The shares are generated such that each party can perform computations, generating a new set of shares that, when reconstructed, are equivalent to performing those exact computations directly on the secret input data. Our variant changes the domain from which secrets are taken: A finite field with prime order is replaced by a geometric algebra over a finite field of prime order. This change preserves the important security properties of Shamir's scheme, namely, idealness (secrets and shares are chosen from the same space) and perfectness ("unauthorized" sets of shares learn nothing about the secret). Our scheme allows secret sharing to be seamlessly added to the arsenal of GA‐based applications. Our extension of Shamir's secret scheme was first worked out for geometric algebras. It appears, however, that in fact it works for other algebras, a situation worthy to be explored in future work. For definiteness, in this paper, we restrict the analysis to the case of geometric algebras. [ABSTRACT FROM AUTHOR]

If an initial frame of vectors {ei}$$ \left\{{e}_i\right\} $$ is related to a final frame of vectors {fi}$$ \left\{{f}_i\right\} $$ by, in geometric algebra (GA) terms, a rotor, or in linear algebra terms, an orthogonal transformation, we often want to find this rotor given the initial and final sets of vectors. One very common example is finding a rotor or orthogonal matrix representing rotation, given knowledge of initial and transformed points. In this paper, we discuss methods in the literature for recovering such rotors and then outline a GA method, which generalises to cases of any signature and any dimension, and which is not restricted to orthonormal sets of vectors. The proof of this technique is both concise and elegant and uses the concept of characteristic multivectors as discussed in the book by Hestenes and Sobczyk, which contains a treatment of linear algebra using geometric algebra. Expressing orthogonal transformations as rotors, enables us to create fractional transformations and we discuss this for some classic transforms. In real applications, our initial and/or final sets of vectors will be noisy. We show how to use the characteristic multivector method to find a 'best fit' rotor between these sets and compare our results with other methods. [ABSTRACT FROM AUTHOR]

Let I$I$ be the ideal of minors of a 2×n$2 \times n$ matrix of linear forms with the expected codimension. In this paper, we prove that the Rees algebra of I$I$ and its special fiber ring are Cohen–Macaulay and Koszul; in particular, they are quadratic algebras. The main novelty in our approach is the analysis of a stratification of the Hilbert scheme of determinantal ideals. We study degenerations of Rees algebras along this stratification, and combine it with certain squarefree Gröbner degenerations. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the local structure of the semi‐invariant picture of a tame hereditary algebra near the null root. Using a construction that we call co‐amalgamation, we show that this local structure is completely described by the semi‐invariant pictures of a collection of self‐injective Nakayama algebras. We then describe the cones of this local structure using cluster‐like structures that we call support regular clusters. Finally, we show that the local structure is (piecewise linearly) invariant under cluster tilting. [ABSTRACT FROM AUTHOR]

A foundational principle in the study of modules over standard graded polynomial rings is that geometric positivity conditions imply vanishing of Betti numbers. The main goal of this paper is to determine the extent to which this principle extends to the nonstandard Z$\mathbb {Z}$‐graded case. In this setting, the classical arguments break down, and the results become much more nuanced. We introduce a new notion of Castelnuovo–Mumford regularity and employ exterior algebra techniques to control the shapes of nonstandard Z$\mathbb {Z}$‐graded minimal free resolutions. Our main result reveals a unique feature in the nonstandard Z$\mathbb {Z}$‐graded case: the possible degrees of the syzygies of a graded module in this setting are controlled not only by its regularity, but also by its depth. As an application of our main result, we show that given a simplicial projective toric variety and a module M$M$ over its coordinate ring, the multigraded Betti numbers of M$M$ are contained in a particular polytope when M$M$ satisfies an appropriate positivity condition. [ABSTRACT FROM AUTHOR]

In this paper we provide some applications of the norm form in some quaternion division algebras over rational field, and we give some properties of Fibonacci sequence and Fibonacci sequence in connection to quaternion elements. We provide some properties of the norm of a rational quaternion algebra, in connection to the famous Lagrange's four‐square theorem and its generalizations given by Ramanujan. Moreover, we prove some results regarding the arithmetic of integer quaternions defined on some division quaternion algebras, and we define and give properties of some special quaternions by using Fibonacci sequences. [ABSTRACT FROM AUTHOR]

Recently, there has been growing interest in developing unmanned aircraft systems (UAS) with advanced onboard autonomous capabilities. This paper describes the current state of the art in autonomous rotorcraft UAS (RUAS) and provides a detailed literature review of the last two decades of active research on RUAS. Three functional technology areas are identified as the core components of an autonomous RUAS. Guidance, navigation, and control (GNC) have received much attention from the research community, and have dominated the UAS literature from the nineties until now. This paper first presents the main research groups involved in the development of GNC systems for RUAS. Then it describes the development of a framework that provides standard definitions and metrics characterizing and measuring the autonomy level of a RUAS using GNC aspects. This framework is intended to facilitate the understanding and the organization of this survey paper, but it can also serve as a common reference for the UAS community. The main objective of this paper is to present a comprehensive survey of RUAS research that captures all seminal works and milestones in each GNC area, with a particular focus on practical methods and technologies that have been demonstrated in flight tests. These algorithms and systems have been classified into different categories and classes based on the autonomy level they provide and the algorithmic approach used. Finally, the paper discusses the RUAS literature in general and highlights challenges that need to be addressed in developing autonomous systems for unmanned rotorcraft. © 2012 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

It has long been known how to read information about the socle degrees of the local cohomology H0m(M) of a graded module over a polynomial ring R from the twists in position d = dimR, in a resolution of M by free R-modules. It has also long been known how to use local cohomology to read valuable information from complexes which approximate resolutions in the sense that they have positive homology of small Krull dimension. The present paper reads information about the maximal generator degree (rather than the socle degree) of H0m(M) from the twists in position d - 1 (rather than position d) in an approximate resolution of M. We apply the local cohomology results to draw conclusions about the maximal generator degree of the second symbolic power of the prime ideal defining a monomial curve and the second symbolic power of the ideal defining a finite set of points in projective space. There is also an application to hyperplane sections of subschemes of projective space and to partial Castelnuovo-Mumford regularity. Perhaps, the most important application is to the study of blow-up algebras and their defining equations. The techniques of the present paper are the main tool used in Kustin, Polini and Ulrich [Algebra Number Theory 11 (2017) 1489-1525] to bound the degrees of these equations and thus to identify them in some cases. [ABSTRACT FROM AUTHOR]

In this paper, a link between oriented CW complexes (that will also be mentioned as configurations) and alternative algebras is studied, determining which configurations are associated with those algebras. Moreover, the isomorphism classes of each two‐dimensional configuration associated with these algebras is analyzed, providing a new method to classify them. In order to complement the theoretical study, two algorithmic methods are implemented: The first one checks if a given algebra is alternative, while the second one constructs and draws the (pseudo)digraph associated with a given alternative algebra. [ABSTRACT FROM AUTHOR]

Two decades ago, Martin andWoodcock made a surprising and prophetic link between statistical mechanics and representation theory. They observed that the decomposition numbers of the blob algebra (that appeared in the context of transfer matrix algebras) are Kazhdan-Lusztig polynomials in type Ã1. In this paper, we take that observation far beyond its original scope. We conjecture that for Ãn there is an equivalence of categories between the characteristic p diagrammatic Hecke category and a 'blob category' that we introduce (using certain quotients of KLR algebras called generalized blob algebras). Using alcove geometry we prove the 'graded degree' part of this equivalence for all n and all prime numbers p. If our conjecture was verified, it would imply that the graded decomposition numbers of the generalized blob algebras in characteristic p give the p-Kazhdan-Lusztig polynomials in type Ãn. We prove this for Ã1, the only case where the p-Kazhdan-Lusztig polynomials are known. This paper relies extensively on color figures. Some references to color may not be meaningful in the printed version, and we refer the reader to the online version which includes the color figures. [ABSTRACT FROM AUTHOR]

This paper aims to present, in a unified manner, algebraic techniques for least squares problem in quaternionic and split quaternionic mechanics. This paper, by means of a complex representation and a real representation of a generalized quaternion matrix, studies generalized quaternion least squares (GQLS) problem, and derives two algebraic methods for solving the GQLS problem. This paper gives not only algebraic techniques for least squares problem over generalized quaternion algebras, but also a unification of algebraic techniques for least squares problem in quaternionic and split quaternionic theory. [ABSTRACT FROM AUTHOR]

In this paper, we develop algorithmic approach to classify nilsoliton metrics on dimension 8. This approach includes finding eigenvalue type of the nilsoliton derivation, the nullity type, the index of the algebra. It can be considered as a continuation of our papers in Abstract and Applied analysis, volume 2013, 1 to 7, (2013), with article ID 871930, and in Journal of Symbolic Computation 50 (2013), 350 ‐ 373. In our previous work, we classified only ordered type, nilsoliton metric Lie algebras ie, the algebras with the derivation type (1  <  2  <  3...  <  n) in dimension 8 and 9. Here, we consider more general case. We consider such metrics with simple derivations on an indecomposable nilpotent Lie algebra. In one of our previous study, we have already classified nilsoliton metric Lie algebras with nonsingular Gram matrix in dimension 8 in Journal of Symbolic Computation, vol: 50, 350 ‐ 373, 2013. Here, we focus on the metrics with singular Gram matrix. We also develop faster algorithm in classifying such metrics. [ABSTRACT FROM AUTHOR]

The components of complex analytic functions define solutions for the Laplace's equation, and in a simply connected domain, each solution of this equation is the first component of a complex analytic function. In this paper, we generalize this result; for each PDE of the form Auxx+Buxy+Cuyy=0, and for each affine planar vector field φ, we give an algebra 픸 with unit e = e1, with respect to which the components of all functions of the form L∘φ are all the solutions for this PDE, where L is differentiable in the sense of Lorch with respect to 픸. Solutions are also constructed for the following equations: Auxx+Buxy+Cuyy+Dux+Euy+Fu=0, 3rd‐order PDEs, and 4th‐order PDEs; among these are the bi‐harmonic, the bi‐wave, and the bi‐telegraph equations. [ABSTRACT FROM AUTHOR]

Quantum B‐algebras are partially ordered algebras characterizing the residuated structure of a quantale. Examples arise in algebraic logic, non‐commutative arithmetic, and quantum theory. A quantum B‐algebra with trivial partial order is equivalent to a group. The paper introduces a corresponding analogue of quantale modules. It is proved that every quantum B‐module admits an injective envelope which is a quantale module. The injective envelope is constructed explicitly as a completion, a multi‐poset version of the completion of Dedekind and MacNeille. [ABSTRACT FROM AUTHOR]

It has been known since (Pillay, J. Pure Appl. Algebra 53 (1988), no. 3, 239–255)that any group definable in an o$o$‐minimal expansion of the real field can be equipped with a Lie group structure. It is therefore natural to ask when is a Lie group Lie isomorphic to a group definable in such an expansion. Conversano, Starchenko and the first author answered this question in (Conversano, Onshuus, and Starchenko, J. Inst. Math. Jussieu 17 (2018), no. 2, 441–452) in the case when the group is solvable. This paper answers similar questions in more general contexts. We first give a complete classification in the case when the group is linear. Specifically, a linear Lie group G$G$ is Lie isomorphic to a group definable in an o$o$‐minimal expansion of the reals if and only if its solvable radical has the same property. We then deal with the general case of a connected Lie group, although unfortunately, we cannot achieve a full characterization. Assuming that a Lie group G$G$ has its Levi subgroups with finite center, we prove that in order for G$G$ to be Lie isomorphic to a definable group it is necessary and sufficient that its solvable radical satisfies the conditions given in (Conversano, Onshuus, and Starchenko, J. Inst. Math. Jussieu 17 (2018), no. 2, 441–452). [ABSTRACT FROM AUTHOR]