Real Algebra Research Articles

Three Algorithms for Grouping Students: A Bridge Between Personalized Tutoring System Data and Classroom Pedagogy

Computer-assisted instructional programs such as intelligent tutoring systems are often used to support blended learning practices in K-12 education, as they aim to meet individual student needs with personalized instruction. While these systems have been shown to be effective under certain conditions, they can be difficult to integrate into pedagogical practices. In this paper, we introduce three group formation algorithms that leverage learning data from the adaptive intelligent tutoring system ALEKS to support pedagogical and collaborative learning practices with ALEKS. Each grouping method was devised for different use cases, but they all utilize a fine-grained multidimensional view of student ability measured across several hundred skills in an academic course. As such, the grouping algorithms not only identify groups of students, but they also determine what areas of ALEKS content each group should focus on. We then evaluate each of the three methods against two alternative baseline methods, which were chosen for their plausibility of being used in practice—one that groups students randomly and one that groups students based on a unidimensional course score. To evaluate these methods, we establish a set of practical metrics based on what we anticipate teachers would care about in practice. Evaluations were performed by simulating mock groupings of students at different time periods for real ALEKS algebra classes that occurred between 2017 and 2019. We show that each devised method obtains more favorable results on the specified metrics than the alternative methods under each use-case. Moreover, we highlight examples where our methods lead to more nuanced groupings than grouping based on a unidimensional measure of ability.

Periodic Clifford symmetry algebras on flux lattices

Real Clifford algebras play a fundamental role in the eight real Altland-Zirnbauer symmetry classes and the classification tables of topological phases. Here, we present another elegant realization of real Clifford algebras in the $d$-dimensional spinless rectangular lattices with $\pi$ flux per plaquette. Due to the $T$-invariant flux configuration, real Clifford algebras are realized as projective symmetry algebras of lattice symmetries. Remarkably, $d$ mod $8$ exactly corresponds to the eight Morita equivalence classes of real Clifford algebras with eightfold Bott periodicity, resembling the eight real Altland-Zirnbauer classes. The representation theory of Clifford algebras determines the degree of degeneracy of band structures, both at generic $k$ points and at high-symmetry points of the Brillouin zone. Particularly, we demonstrate that the large degeneracy at high-symmetry points offers a rich resource for forming novel topological states by various dimerization patterns, including a $3$D higher-order semimetal state with double-charged bulk nodal loops and hinge modes, a $4$D nodal surface semimetal with $3$D surface solid-ball zero modes, and $4$D M\"{o}bius topological insulators with a eightfold surface nodal point or a fourfold surface nodal ring. Our theory can be experimentally realized in artificial crystals by their engineerable $\mathbb{Z}_2$ gauge fields and capability to simulate higher dimensional systems.

On the Component Group of a Real Algebraic Group

On the Component Group of a Real Algebraic Group

Binary Encoded Recursive Generation of Quantum Space-Times

Real geometric algebras distinguish between space and time; complex ones do not. Space-times can be classified in terms of number n of dimensions and metric signature s (number of spatial dimensions minus number of temporal dimensions). Real geometric algebras are periodic in s, but recursive in n. Recursion starts from the basis vectors of either the Euclidean plane or the Minkowskian plane. Although the two planes have different geometries, they have the same real geometric algebra. The direct product of the two planes yields Hestenes’ space-time algebra. Dimensions can be either open (for space-time) or closed (for the electroweak force). Their product yields the eight-fold way of the strong force. After eight dimensions, the pattern of real geometric algebras repeats. This yields a spontaneously expanding space-time lattice with the physics of the Standard Model at each node. Physics being the same at each node implies conservation laws by Noether’s theorem. Conservation laws are not pre-existent; rather, they are consequences of the uniformity of space-time, whose uniformity is a consequence of its recursive generation.

Real Spinors and Real Dirac Equation

We reexamine the minimal coupling procedure in the Hestenes' geometric algebra formulation of the Dirac equation, where spinors are identified with the even elements of the real Clifford algebra of spacetime. This point of view, as we argue, leads naturally to a non-Abelian generalisation of the electromagnetic gauge potential.

Approximate Pythagoras numbers on ⁎-algebras over [formula omitted

The Pythagoras number of a sum of squares is the shortest length among its sums of squares representations. In many algebras, for example real polynomial algebras in two or more variables, there exists no upper bound on the Pythagoras number for all sums of squares. In this paper, we study how Pythagoras numbers in ⁎-algebras over C behave with respect to small perturbations of elements. More precisely, the approximate Pythagoras number of an element is the smallest Pythagoras number among all elements in its ε-ball. We show that these approximate Pythagoras numbers are often significantly smaller than their exact versions, and allow for (almost) dimension-independent upper bounds. Our results use low-rank approximations for Gram matrices of sums of squares and estimates for the operator norm of the Gram map.

A geometrical interpretation of Okubo Spin group

In this work we define, for the first time, the affine and projective plane over the real Okubo algebra, showing a concrete geometrical interpretation of its Spin group. Okubo algebra is a flexible, composition algebra which is also a not unital division algebra. Even though, Okubo algebra has been known for more than 40 years, we believe that this is the first time the algebra was used for affine and projective geometry. After showing that all axioms of affine geometry are verified, we define a projective plane over Okubo algebra as completion of the affine plane and directly through the use of Veronese coordinates. We then present a bijection between the two constructions. Finally we show a geometric interpretation of Spin(O) as the group of collineations that preserve the axis of the plane.

Classification of the Realizations of Lie Algebras for the Vector Fields on a Circle

We describe the realizations of finite-dimensional Lie algebras of smooth tangential vector fields on a circle and construct “canonical” realizations of the two-dimensional noncommutative algebra, as well as the algebra 𝔰𝔩(2,ℝ) . It is shown that any realization of these algebras by smooth vector fields can be reduced to one of “canonical” realizations with the help of piecewise-smooth global transformations of a circle onto itself. We also deduce the formulas for the number of nonequivalent realizations.

Abilov’s estimates for the Clifford–Fourier transform in real Clifford algebras analysis

Abilov’s estimates for the Clifford–Fourier transform in real Clifford algebras analysis

Coordinate‐free exponentials of general multivector in Clp,q algebras for p+q=3

Closed form expressions in real Clifford geometric algebras , and are presented in a coordinate‐free form for exponential function when the exponent is a general multivector. The main difficulty in solving the problem is connected with an entanglement (or mixing) of vector and bivector components and in a form . After disentanglement, the obtained formulas simplify to the well‐known de Moivre‐type trigonometric/hyperbolic function for vector or bivector exponentials. The presented formulas may find wide application in solving GA differential equations, in signal processing, automatic control and robotics.

The Solvability of a System of Quaternion Matrix Equations Involving ϕ-Skew-Hermicity

Let H be the real quaternion algebra and Hm×n denote the set of all m×n matrices over H. For A∈Hm×n, we denote by Aϕ the n×m matrix obtained by applying ϕ entrywise to the transposed matrix AT, where ϕ is a non-standard involution of H. A∈Hn×n is said to be ϕ-skew-Hermicity if A=−Aϕ. In this paper, we provide some necessary and sufficient conditions for the existence of a ϕ-skew-Hermitian solution to the system of quaternion matrix equations with four unknowns AiXi(Ai)ϕ+BiXi+1(Bi)ϕ=Ci,(i=1,2,3),A4X4(A4)ϕ=C4.

Completeness for the Complexity Class forall exists mathbb {R} and Area-Universality

Exhibiting a deep connection between purely geometric problems and real algebra, the complexity class exists mathbb {R} plays a crucial role in the study of geometric problems. Sometimes exists mathbb {R} is referred to as the ‘real analog’ of NP. While NP is a class of computational problems that deals with existentially quantified boolean variables, exists mathbb {R} deals with existentially quantified real variables. In analogy to Pi _2^p and Sigma _2^p in the famous polynomial hierarchy, we study the complexity classes forall exists mathbb {R} and exists forall mathbb {R} with real variables. Our main interest is the AreaUniversality problem, where we are given a plane graph G, and ask if for each assignment of areas to the inner faces of G, there exists a straight-line drawing of G realizing the assigned areas. We conjecture that AreaUniversality is forall exists mathbb {R}-complete and support this conjecture by proving exists mathbb {R}- and forall exists mathbb {R}-completeness of two variants of AreaUniversality. To this end, we introduce tools to prove forall exists mathbb {R}-hardness and membership. Finally, we present geometric problems as candidates for forall exists mathbb {R}-complete problems. These problems have connections to the concepts of imprecision, robustness, and extendability.

Stability of the Planar Quadratic Systems from the Ring-Theoretic Viewpoint

We show that the classical result on the stability of the origin in a quadratic planar system of ODEs can be formulated using either matrix theory or via its associated real and complex Marcus algebra. A generalization to a three-dimensional case is considered and some counterexamples provided.

Scalable spaces

Scalable spaces are simply connected compact manifolds or finite complexes whose real cohomology algebra embeds in their algebra of (flat) differential forms. This is a rational homotopy invariant property and all scalable spaces are formal; indeed, scalability can be thought of as a metric version of formality. They are also characterized by particularly nice behavior from the point of view of quantitative homotopy theory. Among other results, we show that spaces which are formal but not scalable provide counterexamples to Gromov’s long-standing conjecture on distortion in higher homotopy groups.

On the R-matrix realization of quantum loop algebras

We consider rr-matrix realization of the quantum deformations of the loop algebras \tilde{\mathfrak{g}}𝔤̃ corresponding to non-exceptional affine Lie algebras of type \widehat{\mathfrak{g}}=A^{(1)}_{N-1}𝔤̂=AN−1(1), B^{(1)}_nBn(1), C^{(1)}_nCn(1), D^{(1)}_nDn(1), A^{(2)}_{N-1}AN−1(2). For each U_q(\tilde{\mathfrak{g}})Uq(𝔤̃) we investigate the commutation relations between Gauss coordinates of the fundamental \mathbb{L}𝕃-operators using embedding of the smaller algebra into bigger one. The new realization of these algebras in terms of the currents is given. The relations between all off-diagonal Gauss coordinates and certain projections from the ordered products of the currents are presented. These relations are important in applications to the quantum integrable models.

Continuous controlled cone metric-type spaces over real Banach algebras and fixed-point results

In this article, we introduce a new geometrical structure that is the hybrid of a cone metric space over Banach algebra and a controlled metric-type space. We introduce a new metric space and prove analogs of Banach-, Kannan- and Reich-type fixed-point theorems. We also furnish various concrete examples to establish the validity of our results. The obtained results generalize many well-known results in the literature.

Maximal Injective Real W∗-Subalgebras of Tensor Products of Real W∗-Algebras

It is known that injective (complex or real) W∗-algebras with particular factors have been studied well enough. In the arbitrary cases, i.e., in noninjective case, to investigate (up to isomorphism) W∗-algebras is hard enough, in particular, there exist continuum pairwise nonisomorphic noninjective factors of type II. Therefore, it seems interesting to study maximal injective W∗-subalgebras and subfactors. On the other hand, the study of maximal injective W∗-subalgebras and subfactors is also related to the well-known von Neumann’s bicommutant theorem. In the complex case, such subalgebras were investigated by S. Popa, L. Ge, R. Kadison, J. Fang, and J. Shen. In recent years, studies have also begun in the real case. Let us briefly recall the relevance of considering the real case. It is known that in the works of D. Topping and E. Stormer, it was shown that the study of JW-algebras (nonassociative real analogues of von Neumann algebras) of types II and III is essentially reduced to the study of real W∗-algebras of the corresponding type. It turned out that the structure of real W∗-algebras, generally speaking, differs essentially in the complex case. For example, in the finite-dimensional case, in addition to complex and real matrix algebras, quaternions also arise, i.e., matrix algebras over quaternions. In the infinite-dimensional case, it is proved that there exist, up to isomorphism, two real injective factors of typeIIIλ(0<λ<1), and a countable number of pairwise nonisomorphic real injective factors of typeIII0, whose enveloping (complex) W∗-factors are isomorphic, is constructed. It follows from the above that the study of the real analogue of problems in the theory of operator algebras is topical. Moreover, the real analogue is a generalization of the complex case, since the class of real linear operators is much wider than the class of complex linear operators. In this paper, the maximal injective real W∗-subalgebras of real W∗-algebras or real factors are investigated. For real factorsQ⊂R, it is proven that ifQ+iQis a maximal injective W∗-subalgebra inR+iR, thenQalso is a maximal injective real W∗-subalgebra inR. The converse is proved in the case “II1”-factors, that is, it is shown that ifRis a real factor of typeII1, then the maximal injectivity ofQimplies the maximal injectivity ofQ+iQ. Moreover, it is proven that a maximal injective real subfactorQof a real factorRis a maximal injective real W∗-subalgebra inRif and only ifQis irreducible inR, i.e.,Q′∩R=ℝIwhereIis the unit. The “splitting theorem” of Ge-Kadison in the real case is also proven, namely, ifR1is a finite real factor,R2is a finite real W∗-algebra, andRis a real W∗-subalgebra ofR1⊗¯R2containingR1⊗¯ℝI, then there is some real W∗-subalgebraQ2⊂R2such thatR=R1⊗¯Q2. Moreover, it is given some affirmative answers to the question of S. Popa for the real case.

Entropy Treatment of Evolution Algebras.

In this paper, by introducing an entropy of Markov evolution algebras, we treat the isomorphism of S-evolution algebras. A family of Markov evolution algebras is defined through the Hadamard product of structural matrices of non-negative real S-evolution algebras, and their isomorphism is studied by means of their entropy. Furthermore, the isomorphism of S-evolution algebras is treated using the concept of relative entropy.

The Derived-Discrete Algebras Over the Real Numbers

We classify derived-discrete algebras over the real numbers up to Morita equivalence, using the classification of complex derived-discrete algebras in D. Vossieck, (J. Algebra, 243, 168–176 2001). To this end, we investigate the quiver presentation of the complexified algebra of a real algebra given by a modulated quiver and an admissible ideal.

Local and 2-local derivations on octonion algebras

This paper is devoted to study of local and 2-local derivations on octonion algebras. We shall give a general form of local derivations on the real octonion algebra [Formula: see text]. This description implies that the space of all local derivations on [Formula: see text] when equipped with Lie bracket is isomorphic to the Lie algebra [Formula: see text] of all real skew-symmetric [Formula: see text]-matrices. We also consider [Formula: see text]-local derivations on an octonion algebra [Formula: see text] over an algebraically closed field [Formula: see text] of characteristic zero and prove that every [Formula: see text]-local derivation on [Formula: see text] is a derivation. Further, we apply these results to similar problems for the simple seven-dimensional Malcev algebra. As a corollary, we obtain that the real octonion algebra [Formula: see text] and Malcev algebra [Formula: see text] are simple non-associative algebras which admit pure local derivations, that is, local derivations which are not derivation.