Basic Algebra Research Articles

Arnold’s Piecewise Linear Filtrations, Analogues of Stanley–Reisner Rings and Simplicial Newton Polyhedra

In 1974, the author proved that the codimension of the ideal (g1,g2,…,gd) generated in the group algebra K[Zd] over a field K of characteristic 0 by generic Laurent polynomials having the same Newton polytope Γ is equal to d!×Volume(Γ). Assuming that Newtons polytope is simplicial and super-convenient (that is, containing some neighborhood of the origin), the author strengthens the 1974 result by explicitly specifying the set Bsh of monomials of cardinality d!×Volume(Γ), whose equivalence classes form a basis of the quotient algebra K[Zd]/(g1,g2,…,gd). The set Bsh is constructed inductively from any shelling sh of the polytope Γ. Using the Bsh structure, we prove that the associated graded K -algebra grΓ(K[Zd]) constructed from the Arnold–Newton filtration of K -algebra K[Zd] has the Cohen–Macaulay property. This proof is a generalization of B. Kind and P. Kleinschmitt’s 1979 proof that Stanley–Reisner rings of simplicial complexes admitting shelling are Cohen–Macaulay. Finally, we prove that for generic Laurent polynomials (f1,f2,…,fd) with the same Newton polytope Γ, the set Bsh defines a monomial basis of the quotient algebra K[Zd]/(g1,g2,…,gd).

Applying the Concepts of “Community” and “Social Interaction” from Vygotsky’s Sociocultural Theory of Cognitive Development in Math Teaching to Develop Learner’s Math Communication Competencies

As an integral skill in Math learning at high schools, Math communication competencies are formed and developed throughout the process of Math learning in the classroom environment through student-teacher, student-student as well as student-learning material and instrument interactions. Vygotsky’s Sociocultural Theory of Cognitive Development acts as the foundation and guideline for the teaching and learning process at school, emphasizing the role of social interaction in cognitive development. The study presents the two concepts of ‘community’ and ‘social interactions’ from the theory by Vygotsky for application in Math teaching. The article starts with clarifying the two concepts above in the context of Math classrooms, then proposes several sample activities of teaching Algebra in high schools with the application of the two concepts to develop learners’ Math communication competence.

Landau–Ginzburg mirror, quantum differential equations and qKZ difference equations for a partial flag variety

We consider the system of quantum differential equations for a partial flag variety and construct a basis of solutions in the form of multidimensional hypergeometric functions, that is, we construct a Landau–Ginzburg mirror for that partial flag variety. In our construction, the solutions are labeled by elements of the K-theory algebra of the partial flag variety.To establish these facts we consider the equivariant quantum differential equations for a partial flag variety and introduce a compatible system of difference equations, which we call the qKZ equations. We construct a basis of solutions of the joint system of the equivariant quantum differential equations and qKZ difference equations in the form of multidimensional hypergeometric functions. Then the facts about the non-equivariant quantum differential equations are obtained from the facts about the equivariant quantum differential equations by a suitable limit.Analyzing these constructions we obtain a formula for the fundamental Levelt solution of the quantum differential equations for a partial flag variety.

Simplified Subgrid multiscale stabilized finite element method in the transient framework for Stokes equations

This article presents a novel stabilized finite element analysis for the transient Stokes model. The algebraic subgrid variational multiscale finite element scheme with dynamic subscales approach has been employed to arrive at the stabilized formulation. Both the coarse and the fine scale solutions are of time dependent nature and the unknown fine scale solution is completely eliminated in terms of the coarse scale solution during the derivation. This elimination results into the emergence of a new subgrid multiscale stabilized formulation in the transient framework. This formulation facilitates the theoretical derivations of the robustness properties of the scheme. The fully implicit backward Euler scheme has been applied for the time discretization. Here we have analysed the stability property of the approximate solution. As well as a detailed derivation of the a posteriori error estimate has been presented. The scheme is numerically validated for a benchmark problem and appropriate numerical experiments have been carried out to verify the theoretically established order of convergence results.

Strong Positivity for the Skein Algebras of the 4-Punctured Sphere and of the 1-Punctured Torus

The Kauffman bracket skein algebra is a quantization of the algebra of regular functions on the $${\textit{SL}}_2$$ character variety of a topological surface. We realize the skein algebra of the 4-punctured sphere as the output of a mirror symmetry construction based on higher genus Gromov–Witten theory and applied to a complex cubic surface. Using this result, we prove the positivity of the structure constants of the bracelets basis for the skein algebras of the 4-punctured sphere and of the 1-punctured torus. This connection between topology of the 4-punctured sphere and enumerative geometry of curves in cubic surfaces is a mathematical manifestation of the existence of dual descriptions in string/M-theory for the $$\mathcal {N}=2$$ $$N_f=4$$ SU(2) gauge theory.

An efficient geometric method for incompressible hydrodynamics on the sphere

We present an efficient and highly scalable geometric numerical method for two-dimensional ideal fluid dynamics on the sphere. The starting point is Zeitlin's finite-dimensional model of hydrodynamics. The efficiency stems from exploiting a tridiagonal splitting of the discrete spherical Laplacian combined with highly optimized, scalable numerical algorithms. For time-stepping, we adopt a recently developed isospectral integrator able to preserve the geometric structure of Euler's equations, in particular conservation of the Casimir functions. To overcome previous computational bottlenecks, we formulate the matrix Lie algebra basis through a sequence of tridiagonal eigenvalue problems, efficiently solved by well-established linear algebra libraries. The same tridiagonal splitting allows for computation of the stream matrix, involving the inverse Laplacian, for which we design an efficient parallel implementation on distributed memory systems. The resulting overall computational complexity is O(N3) per time-step for N2 spatial degrees of freedom. The dominating computational cost is matrix-matrix multiplication, carried out via the parallel library ScaLAPACK. Scaling tests show approximately linear scaling up to around 2500 cores for the matrix size N=4096 with a computational time per time-step of about 0.55 seconds. These results allow for long-time simulations and the gathering of statistical quantities while simultaneously conserving the Casimir functions. We illustrate the developed algorithm for Euler's equations at the resolution N=2048.

Hammocks to visualize the support of finitely presented functors

Many properties of a module can be expressed in terms of the dimension of the vector space obtained by applying a finitely presented functor to that module. For example, the dimension of the kernel, image or cokernel of the multiplication map given by an algebra element; or the number of summands of a certain type when the module is considered as a module over a subalgebra. When the indecomposable modules over the algebra are arranged in the Auslander-Reiten quiver, the support of the finitely presented functor typically has the shape of a hammock, spanned between sources and sinks. There may also be tangents which are meshes where the hammock function at the middle term exceeds the sum of the values at the start and end terms. We describe how sources, sinks and tangents of the hammock relate to the modules which define the projective resolution of the finitely presented functor. The key tool is the Cokernel Complex Lemma which links the values of the hammock function to the Auslander-Reiten structure of the category. We are also interested in exact subcategories of module categories which have Auslander-Reiten sequences. Our examples include quiver representations and invariant subspaces of nilpotent linear operators.

Lesson on the theme "Elements of circuitry. Logic diagrams"

The article presents one of the lessons of the "Elements of algebra of logic" section of the informatics course for 10th grade. The content of the lesson corresponds to the teaching materials on informatics by L. L. Bosova, A. Yu. Bosova. The lesson studies such concepts as a logical element (gate), a logical device, a logical circuit, an inverter, a conjunctor, a disjunctor. A feature of the lesson is the use of thematic videos (with and without sound) for viewing in order to study new material and for self-voicing. Organized frontal, independent and group work is aimed at studying the ways of compiling logical formulas for logical circuits and compiling logical circuits for given logical formulas. During the lesson, cadets draw up a summary on the topic and carry out self-assessment using a checklist.

A decidable theory involving addition of differentiable real functions

This paper enriches a pre-existing decision algorithm, which in turn augmented a fragment of Tarski's elementary algebra with one-argument real functions endowed with a continuous first derivative. In its present (still quantifier-free) version, our decidable language embodies the addition of functions and multiplication of functions by scalars; the issue we address is the one of satisfiability.As regards real numbers, individual variables and constructs designating the basic arithmetic operations are available, along with comparison relators. As regards functions, we have variables of another sort, out of which compound terms are formed by means of constructs designating addition and differentiation. An array of predicates designates various relationships between functions, as well as function properties, that may hold over intervals of the real line; those are: function comparisons, strict and non-strict monotonicity / convexity / concavity, comparisons between the derivative of a function and a real-valued term.Our decision method consists in preprocessing the given formula into an equi-satisfiable quantifier-free formula of the elementary algebra of real numbers, whose satisfiability can then be checked by means of Tarski's decision method. No direct reference to functions will appear in the target formula, each function variable having been superseded by a collection of stub real variables; hence, in order to prove that the proposed translation is satisfiability-preserving, we must figure out a flexible-enough family of interpolating C1 functions that can accommodate a model for the source formula whenever the target formula turns out to be satisfiable.With respect to the results announced in earlier papers of the same stream, a significant effort went into designing the family of interpolating functions so that it could meet the new constraints stemming from the presence of function addition (along with differentiation) among the constructs of our fragment of mathematical analysis.

TOD

Outlier detection (OD) is a key machine learning task for finding rare and deviant data samples, with many time-critical applications such as fraud detection and intrusion detection. In this work, we propose TOD, the first tensor-based system for efficient and scalable outlier detection on distributed multi-GPU machines. A key idea behind TOD is decomposing complex OD applications into a small collection of basic tensor algebra operators. This decomposition enables TOD to accelerate OD computations by leveraging recent advances in deep learning infrastructure in both hardware and software. Moreover, to deploy memory-intensive OD applications on modern GPUs with limited on-device memory, we introduce two key techniques. First, provable quantization speeds up OD computations and reduces its memory footprint by automatically performing specific floating-point operations in lower precision while provably guaranteeing no accuracy loss. Second, to exploit the aggregated compute resources and memory capacity of multiple GPUs, we introduce automatic batching , which decomposes OD computations into small batches for both sequential execution on a single GPU and parallel execution across multiple GPUs. TOD supports a diverse set of OD algorithms. Evaluation on 11 real-world and 3 synthetic OD datasets shows that TOD is on average 10.9X faster than the leading CPU-based OD system PyOD (with a maximum speedup of 38.9X), and can handle much larger datasets than existing GPU-based OD systems. In addition, TOD allows easy integration of new OD operators, enabling fast prototyping of emerging and yet-to-discovered OD algorithms.

Graded extension of Thomas-Whitehead gravity

Thomas-Whitehead (TW) gravity was recently introduced as a projective gauge theory of gravity over a d-dimensional manifold that embeds reparametrization invariance into the action functional for gravitation through the use of the Thomas-Whitehead connection. The projective invariance in this d-dimensional theory enjoys an intimate relationship with the Virasoro coadjoint elements found in string theory as one of the components of the connection, ${\mathcal{D}}_{ab}$, is directly related to the coadjoint elements of the Virasoro algebra. TW gravity exploits projective Gauss-Bonnet terms in the action functional which allows the theory to collapse to Einstein's theory of general relativity in the limit that ${\mathcal{D}}_{ab}$ vanishes. In this paper we develop the graded extension of TW gravity, super-TW gravity, in the framework of a DeWitt supermanifold. We construct the Lagrangian for super-TW gravity, give a detailed derivation of the classical field equations, and discuss the graded extension of the projective connection as a prelude to a future understanding of TW-supergravity (which has manifest supersymmetry) and its relationship to the super-Virasoro algebra.

Spectra of algebras of block-symmetric analytic functions of bounded type

We investigate algebras of block-symmetric analytic functions on spaces $\ell_{p}(\mathbb{C}^s)$ which are $\ell_{p}$-sums of $\mathbb{C}^{s}.$ We consider properties of algebraic bases of block-symmetric polynomials,intertwining operations on spectra of the algebras and representations of the spectra as a semigroup of analytic functions of exponential type of several variables. All invertible elements of the semigroup are described for the case $p=1.$

A simplified treatment of Ramana’s exact dual for semidefinite programming

In semidefinite programming the dual may fail to attain its optimal value and there could be a duality gap, i.e., the primal and dual optimal values may differ. In a striking paper, Ramana (Math. Program. Ser. B 77, 129–162, 1997) proposed a polynomial size extended dual that does not have these deficiencies and yields a number of fundamental results in complexity theory. In this work we walk the reader through a concise and self-contained derivation of Ramana’s dual, relying mostly on elementary linear algebra.

Providing performance portable numerics for Intel GPUs

SummaryWith discrete Intel GPUs entering the high‐performance computing landscape, there is an urgent need for production‐ready software stacks for these platforms. In this article, we report how we enable the Ginkgo math library to execute on Intel GPUs by developing a kernel backed based on the DPC++ programming environment. We discuss conceptual differences between the CUDA and DPC++ programming models and describe workflows for simplified code conversion. We evaluate the performance of basic and advanced sparse linear algebra routines available in Ginkgo's DPC++ backend in the hardware‐specific performance bounds and compare against routines providing the same functionality that ship with Intel's oneMKL vendor library.

Sandwich Theorems for a New Class of Complete Homogeneous Symmetric Functions by Using Cyclic Operator

In this paper, we discuss and introduce a new study on the connection between geometric function theory, especially sandwich theorems, and Viete’s theorem in elementary algebra. We obtain some conclusions for differential subordination and superordination for a new formula of complete homogeneous symmetric functions class involving an ordered cyclic operator. In addition, certain sandwich theorems are found.

Non-abelian extensions of pre-Lie algebras

In this paper, we study non-abelian extensions of pre-Lie algebras. First we introduce the non-abelian cohomology for pre-Lie algebras by which we classify non-abelian extensions of pre-Lie algebras. Then we construct the bimultipliers for pre-Lie algebras, which naturally gives rise to a strict Lie 2-algebra. We show that there is a one-to-one correspondence between isomorphic classes of non-abelian extensions of pre-Lie algebras and homotopy classes of homomorphisms from the subadjacent Lie algebra to the strict Lie 2-algebra obtained from the bimultipliers. Finally we classify non-abelian extensions of pre-Lie algebras by Maurer-Cartan elements of a differential graded Lie algebra.

Algebraic construction of Weyl invariant E8 Jacobi forms

We study the ring of Weyl invariant E8 weak Jacobi forms. Wang recently proved that the ring is not a polynomial algebra. We consider a polynomial algebra which contains the ring as a subset and clarify the necessary and sufficient condition for an element of the polynomial algebra to be a Weyl invariant E8 weak Jacobi form. This serves as a new algorithm for constructing all the Jacobi forms of given weight and index. The algorithm is purely algebraic and does not require Fourier expansion. Using this algorithm we determine the generators of the free module of Weyl invariant E8 weak Jacobi forms of given index m for m≤20. We also determine the lowest weight generators of the free module of index m for m≤28. Our results support the lower bound conjecture of Sun and Wang and prove explicitly that there exist generators of the ring of Weyl invariant E8 weak Jacobi forms of weight −4m and index m for all 12≤m≤28.

Why Quaternions and Octonions Exist

Summary There is a simple combinatorial anomaly, making possible some special linear algebra and thereby some special geometry, that occurs only in dimensions 1, 2, 4, and 8. The consequences are wide ranging and in particular lead to the existence of the complex numbers, the quaternions and the octonions. This article explains why the anomaly exists only in these dimensions using elementary linear algebra.

From the historical text to the classroom session: analysing the work of teachers-as-designers.

While classical studies have highlighted the many potential benefits of using original historical sources in the classroom, few studies have documented actual classroom practices outside of research contexts. In this case study, I aim to describe and explain how five French high school teachers autonomously designed and implemented classroom sessions starting from the same document, namely an excerpt from Euler’s Elements of Algebra presenting an algorithm for square root approximation. From a methodological viewpoint, it enables me show how two general frameworks for the study of teachers’ professional practices—the Documentational Approach to Didactics and the Didactic and Ergonomic Double Approach—can be tailored to fit the specific challenges of using historical sources. The empirical results provide fresh insights into the conditions for a mathematically rich use of historical sources in the classroom, and on the connections between this use and the integration of a historical perspective in the teaching of mathematics.

Lie symmetry analysis for two-phase flow with mass transfer

Abstract We perform a complete symmetry classification for the hyperbolic system of partial differential equations, which describes a drift-flux two-phase flow in a one-dimensional pipe, with a mass-transfer term between the two different phases of the fluid. In addition, we consider the polytropic equation of states parameter and gravitational forces. For general values of the polytropic indices, we find that the fluid equations are invariant under the elements of a three-dimensional Lie algebra. However, additional Lie point symmetries follow for specific values of the polytropic indices. The one-dimensional systems are investigated in each case of the classification scheme, and the similarity transformations are calculated in order to reduce the fluid equations into a system of ordinary differential equations. Exact solutions are derived, while the reduced systems are studied numerically.