Real Algebra Research Articles

On semireal hyperrings and real hyperrings

The main aim of this paper is to develop the study of hyperrings in real algebra. By introducing the notions of real hyperrings and semireal hyperrings, we obtain some necessary and sufficient conditions for real (semireal) hyperrings. Also, we characterize the minimal prime hyperideals in hyperrings.

Polynomial realizations of finite-dimensional Lie algebras

Доказано, что произвольная конечномерная алгебра Ли над алгебраически замкнутым полем $K$ характеристики $0$ может быть вложена (реализована) в качестве транзитивной подалгебры Ли в алгебру Ли полиномиальных векторных полей на пространстве $K^n$. То же доказано для произвольных вещественных алгебр Ли и еще в некоторых других случаях.

Some classes of four-dimensional real division algebras with few automorphisms

Some classes of four-dimensional real division algebras with few automorphisms

Algebras generated by special symmetric polynomials on $\ell_1$

Let $X$ be a weighted direct sum of infinity many copies of complex spaces $\ell_1\bigoplus \ell_1.$ We consider an algebra consisting of polynomials on $X$ which are supersymmetric on each term $\ell_1\bigoplus \ell_1.$ Point evaluation functionals on such algebra gives us a relation of equivalence `$\sim$' on $X.$ We investigate the quotient set $X/\sim$ and show that under some conditions, it has a real topological algebra structure.

Pattern groups and a poset based Hopf monoid

The supercharacter theory of algebra groups gave us a representation theoretic realization of the Hopf algebra of symmetric functions in noncommuting variables. The underlying representation theoretic framework comes equipped with two canonical bases, one of which was completely new in terms of symmetric functions. This paper simultaneously generalizes this Hopf structure by considering a larger class of groups while also restricting the representation theory to a more combinatorially tractable one. Using the normal lattice supercharacter theory of pattern groups, we not only gain a third canonical basis, but also are able to compute numerous structure constants in the corresponding Hopf monoid, including coproducts and antipodes for the new bases.

Three-Partition Hodge Integrals and the Topological Vertex

A conjecture on the relation between the cubic Hodge integrals and the topological vertex in topological string theory is resolved. A central role is played by the notion of generalized shift symmetries in a fermionic realization of the two-dimensional quantum torus algebra. These algebraic relations of operators in the fermionic Fock space are used to convert generating functions of the cubic Hodge integrals and the topological vertex to each other. As a byproduct, the generating function of the cubic Hodge integrals at special values of the parameters therein is shown to be a tau function of the generalized KdV (aka Gelfand-Dickey) hierarchies.

IDENTIFIKASI INDIKATOR KEMAMPUAN KOMUNIKASI MATEMATIS PADA SOAL USBN MATEMATIKA SD/MI TAHUN AJARAN 2018/2019

Mathematic communication skills is one of the critical indicator in learning and evaluation of mathematic lesson. But in reality, not all tests accepted by students can upgrade their mathematic communication skills. Meanwhile based on Regulation from National Education Minister Number 22 Year of 2006 it is mentioned that one of the goals of mathematic learning is for students have capability to communicate ideas with symbols, tables, charts, and any other media to clarify the conditions or problems. Based on that it is necessary to identify mathematic communication skills indicator on mathematic tests. Moreover on the mathematic USBN, where the results determine the students’s final score. The mathematic communication skills indicator which would be identified were: 1) connects the real things, picture and charts to the mathematic ideas, 2) explains ideas, situations and mathematic relations with speech or texts with real things, pictures, graphs and algebra, 3) listens, discusses and writes about mathematics, 4) states daily activity in mathematic languace, 5) reads with understanding about written mathematic presentation, 6) makes mathematic statement which is relevant with problem situation, and 7) makes conjectures, arguments, determines definition and generalisation. In here the mathematic communication skills that would be analyzed was students capability to solve contextual problems. On Mathematic .USBN for Elementary School in the year 2018/2019 there were 35 problems, where 30 problems were multiple choice and 5 were essay. There were few problems contains mathematic communication skills indicator which were number 31, 32, 33, 34, 35

On the Construction of Converging Hierarchies for Polynomial Optimization Based on Certificates of Global Positivity

In recent years, techniques based on convex optimization and real algebra that produce converging hierarchies of lower bounds for polynomial minimization problems have gained much popularity. At their heart, these hierarchies rely crucially on Positivstellensätze from the late 20th century (e.g., due to Stengle, Putinar, or Schmüdgen) that certify positivity of a polynomial on an arbitrary closed basic semialgebraic set. In this paper, we show that such hierarchies could in fact be designed from much more limited Positivstellensätze dating back to the early 20th century that only certify positivity of a polynomial globally. More precisely, we show that any inner approximation to the cone of positive homogeneous polynomials that is arbitrarily tight can be turned into a converging hierarchy of lower bounds for general polynomial minimization problems with compact feasible sets. This in particular leads to a semidefinite programming–based hierarchy that relies solely on Artin’s solution to Hilbert’s 17th problem. We also use a classical result from Pólya on global positivity of even forms to construct an “optimization-free” converging hierarchy for general polynomial minimization problems with compact feasible sets. This hierarchy requires only polynomial multiplication and checking nonnegativity of coefficients of certain fixed polynomials. As a corollary, we obtain new linear programming–based and second-order cone programming–based hierarchies for polynomial minimization problems that rely on the recently introduced concepts of diagonally dominant sum of squares and scaled diagonally dominant sum of squares polynomials. We remark that the scope of this paper is theoretical at this stage, as our hierarchies—though they involve at most two sum of squares constraints or only elementary arithmetic at each level—require the use of bisection and increase the number of variables (respectively, the degree) of the problem by the number of inequality constraints plus three (respectively, by a factor of two).

A Note of The Linear Equation AX = B with Multiplicatively-Reguler Matrix A in Semiring

Semiring is a form of generalization of the ring, where one or more conditions in the ring are removed. An element a is called multiplicatively-regular if there is x so axa = a. In real number algebra, a system of linear equations AX = B has a singular solution if a matrix A has an inverse. Elements of semiring which does not a zero element have no inverse of addition. By reviewing matrix A as a multiplicatively-regular, it is develop of necessary or sufficient condition of semiring. Given a matrix A with the right complement matrix A r satisfies AA r = 0. The sufficient condition of the linear equations system AX = B has a solution is there exist a matrix B satisfies AA [openbullet] B = B and a matrix A has a right complement matrix A r.

Realizations of the Witt and Virasoro Algebras and Integrable Equations

In this paper we study realizations of infinite-dimensional Witt and Virasoro algebras. We obtain a complete description of realizations of the Witt algebra by Lie vector fields of first-order differential operators over the space ℝ3. We prove that none of them admits non-trivial central extension, which means that there are no realizations of the Virasoro algebra in ℝ3. We describe all inequivalent realizations of the direct sum of the Witt algebras by Lie vector fields over ℝ3. This result enables complete description of all possible (1+1)-dimensional partial differential equations that admit infinite dimensional symmetry algebras isomorphic to the direct sum of Witt algebras. In this way we have constructed a number of new classes of nonlinear partial differential equations admitting infinite-dimensional Witt algebras. So new integrable models which admit infinite symmetry algebra are obtained.

A broad class of discrete-time hypercomplex-valued Hopfield neural networks

In this paper, we address the stability of a broad class of discrete-time hypercomplex-valued Hopfield-type neural networks. To ensure the neural networks belonging to this class always settle down at a stationary state, we introduce novel hypercomplex number systems referred to as real-part associative hypercomplex number systems. Real-part associative hypercomplex number systems generalize the well-known Cayley–Dickson algebras and real Clifford algebras and include the systems of real numbers, complex numbers, dual numbers, hyperbolic numbers, quaternions, tessarines, and octonions as particular instances. Apart from the novel hypercomplex number systems, we introduce a family of hypercomplex-valued activation functions called B-projection functions. Broadly speaking, a B-projection function projects the activation potential onto the set of all possible states of a hypercomplex-valued neuron. Using the theory presented in this paper, we confirm the stability analysis of several discrete-time hypercomplex-valued Hopfield-type neural networks from the literature. Moreover, we introduce and provide the stability analysis of a general class of Hopfield-type neural networks on Cayley–Dickson algebras.

Donoho–Stark’s Uncertainty Principles in Real Clifford Algebras

The Clifford Fourier transform (CFT) has been shown to be a powerful tool in the Clifford analysis. In this work, several uncertainty inequalities are established in the real Clifford algebra $Cl_{(p,q)}$, \ including the Hausdorf-Young inequality, and three qualitative uncertainty principles of Donoho-Stark.

Using Geometric Interval Algebra Modeling for Improved Three-Dimensional Camera Calibration

This paper addresses the problem of estimating camera calibration parameters by using a novel method based on interval algebra. Unlike existing solutions, which usually apply real algebra, our method is capable of obtaining highly accurate parameters even in scenarios where the input data for camera calibration are severely corrupted by noise or no artificial calibration target can be introduced on the scene. We introduce some key concepts regarding the usage of interval algebra on projective space, which might be used by other computer vision methods. To demonstrate the robustness and effectiveness of our method, we present results for camera calibration with varying levels of noise on the input data of a world coordinate frame (standard deviation of up to 0.5 m) and their corresponding projections onto an image plane (standard deviation of up to 10 pixels), which are significantly larger than noise levels considered by state-of-the-art methods.

Boundaries for the Cartesian product of real function algebras

The concepts of Choquet sets and Silov boundary for a real function algebra and real function space has been introduced and studied mainly by Kulkarni and Limaye. Here we study these concepts for the Cartesian product of real function algebras and function spaces.

A note on q-oscillator realizations of Uq(gl(M|N)) for Baxter Q-operators

We consider asymptotic limits of q-oscillator (or Heisenberg) realizations of Verma modules over the quantum superalgebra Uq(gl(M|N)), and obtain q-oscillator realizations of the contracted algebras proposed in [1]. Instead of factoring out the invariant subspaces, we make reduction on generators of the q-oscillator algebra, which gives a shortcut to the problem. Based on this result, we obtain explicit q-oscillator representations of a Borel subalgebra of the quantum affine superalgebra Uq(glˆ(M|N)) for Baxter Q-operators.

A Discrete Realization of the Higher Rank Racah Algebra

In previous work, a higher rank generalization R(n) of the Racah algebra was defined abstractly. The special case of rank one encodes the bispectrality of the univariate Racah polynomials and is known to admit an explicit realization in terms of the operators associated with these polynomials. Starting from the Dunkl model for which we have an action by R(n) on the Dunkl-harmonics, we show that connection coefficients between bases of Dunkl-harmonics diagonalizing certain Abelian subalgebras are multivariate Racah polynomials. By lifting the action of R(n) to the connection coefficients, we identify the action of the Abelian subalgebras with the action of the Racah operators defined by J. S. Geronimo and P. Iliev. Making appropriate changes of basis, one can identify each generator of R(n) as a discrete operator acting on the multivariate Racah polynomials.

Dual Numbers and Operational Umbral Methods

Dual numbers and their higher-order version are important tools for numerical computations, and in particular for finite difference calculus. Based on the relevant algebraic rules and matrix realizations of dual numbers, we present a novel point of view, embedding dual numbers within a formalism reminiscent of operational umbral calculus.

Affine cluster monomials are generalized minors

We study the realization of acyclic cluster algebras as coordinate rings of Coxeter double Bruhat cells in Kac–Moody groups. We prove that all cluster monomials with$\mathbf{g}$-vector lying in the doubled Cambrian fan are restrictions of principal generalized minors. As a corollary, cluster algebras of finite and affine type admit a complete and non-recursive description via (ind-)algebraic group representations, in a way similar in spirit to the Caldero–Chapoton description via quiver representations. In type$A_{1}^{(1)}$, we further show that elements of several canonical bases (generic, triangular, and theta) which complete the partial basis of cluster monomials are composed entirely of restrictions of minors. The discrepancy among these bases is accounted for by continuous parameters appearing in the classification of irreducible level-zero representations of affine Lie groups. We discuss how our results illuminate certain parallels between the classification of representations of finite-dimensional algebras and of integrable weight representations of Kac–Moody algebras.

Light-front bootstrap for Chern-Simons matter theories

We propose a new approach to solve conformal field theories and apply it to Chern-Simons Matter theories and three-dimensional bosonization duality. All three-point correlation functions of single-trace operators are obtained in the large-N as a simple application. The idea is to construct, as an effective weakly-coupled theory, a nonlinear realization of the conformal algebra in terms of physical, gauge-invariant, operators. The efficiency of the method is also in the use of an analog of the light-cone gauge and of the momentum-space on the CFT side. AdS/CFT is used as a convenient regulator and as a source of the canonical bracket. The uniqueness of the nonlinear realization manifests the three-dimensional bosonization duality at this order. We also find two more non-unitary solutions which should be analogous to the fishnet theories. The results can also be viewed as an explicit realization of the slightly-broken higher spin symmetry.As a by-product, the cubic action of the Higher Spin Gravity in AdS4 is constructed. While generic Higher Spin Gravities are obstructed at higher orders by nonlocality, we point out the existence of two especially simple and well-defined theories: chiral and anti-chiral whose three-point functions correspond to the two new solutions. These two theories are supposed to give a quantum complete and local example of gravitational bulk duals.

Are octonions necessary to the Standard Model?

There have been a number of claims, going back to the 1970s, that the Standard Model of particle physics, based on fermions and antifermions, might be derived from an octonion algebra. The emergence of SU(3), SU(2) and U(1) groups in octonion-based structures is suggestive of the symmetries of the Standard Model, but octonions themselves are an unsatisfactory model for physical application because they are antiassociative and consequently not a group. Instead, the ‘octonion’ models have to be based on adjoint algebras, such as left - or right-multiplied octonions, which can be seen to have group-like properties. The most promising of these candidates is the complexified left-multiplied octonion algebra, because it reduces, in effect, to Cl(6), which has been identified by one of us (PR) in a number of previous publications as the basic structure for the entire foundation of physics, as well as the algebra required for the Standard Model and the Dirac equation. Though this algebra has long been shown by PR as equivalent to using a complexified left-multiplied or ‘broken’ octonion, it doesn’t need to be derived in this way, as its real origins are in the respective real, complex, quaternion and complexified quaternion algebras of the fundamental parameters of mass, time, charge and space. The ‘broken’ octonion, however, does have value in leading to the higher (and equally broken) symmetries, such as E8, which incorporate fermions, with their two spin states, along with gauge bosons and vacuum states into a unified scheme.