Representations Of Algebras Research Articles

Parameter learning of delayed Boolean control networks with missing observations

In this paper, a new and effective parameter learning method for delayed Boolean control networks (DBCNs) with missing observations is presented in this paper, which can handle strong noise and converge to the true parameter values in finite times. Two types of noise that follow different time-varying Bernoulli distributions are considered in this paper: measurement noise and system noise. The concept of virtual nodes is introduced by the paper to cope with random missing observations, and the algebraic representation of DBCNs is obtained using the STP method. A parameter learning algorithm based on the forward and backward probability and EM algorithm is then proposed by the paper, which can estimate the model parameters from time series data. The performance and robustness of the proposed method are demonstrated by the paper through a numerical example.

Proving Query Equivalence Using Linear Integer Arithmetic

Proving the equivalence between SQL queries is a fundamental problem in database research. Existing solvers model queries using algebraic representations and convert such representations into first-order logic formulas so that query equivalence can be verified by solving a satisfiability problem. The main challenge lies in "unbounded summations", which appear commonly in a query's algebraic representation in order to model common SQL features, such as projection and aggregate functions. Unfortunately, existing solvers handle unbounded summations in an ad-hoc manner based on heuristics or syntax comparison, which severely limits the set of queries that can be supported. This paper develops a new SQL equivalence prover called SQLSolver, which can handle unbounded summations in a principled way. Our key insight is to use the theory of LIA^*, which extends linear integer arithmetic formulas with unbounded sums and provides algorithms to translate a LIA^* formula to a LIA formula that can be decided using existing SMT solvers. We augment the basic LIA^* theory to handle several complex scenarios (such as nested unbounded summations) that arise from modeling real-world queries. We evaluate SQLSolver with 359 equivalent query pairs derived from the SQL rewrite rules in Calcite and Spark SQL. SQLSolver successfully proves 346 pairs of them, which significantly outperforms existing provers.

Matrix representation of finite effect algebras

Matrix representation of finite effect algebras

Bosonic Representation of Matrices and Angular Momentum Probabilistic Representation of Cyclic States.

The Jordan-Schwinger map allows us to go from a matrix representation of any arbitrary Lie algebra to an oscillator (bosonic) representation. We show that any Lie algebra can be considered for this map by expressing the algebra generators in terms of the oscillator creation and annihilation operators acting in the Hilbert space of quantum oscillator states. Then, to describe quantum states in the probability representation of quantum oscillator states, we express their density operators in terms of conditional probability distributions (symplectic tomograms) or Husimi-like probability distributions. We illustrate this general scheme by examples of qubit states (spin-1/2 su(2)-group states) and even and odd Schrödinger cat states related to the other representation of su(2)-algebra (spin-j representation). The two-mode coherent-state superpositions associated with cyclic groups are studied, using the Jordan-Schwinger map. This map allows us to visualize and compare different properties of the mentioned states. For this, the su(2) coherent states for different angular momenta j are used to define a Husimi-like Q representation. Some properties of these states are explicitly presented for the cyclic groups C2 and C3. Also, their use in quantum information and computing is mentioned.

Energies of T-spherical fuzzy graph based on novel Aczel-Alsina T-norm and T-conorm with their applications in decision making

A T-spherical fuzzy set (TSFS) is an extended and logical algebraic representation to handle uncertainty, with the help of four functions describing four possible aspects of uncertain information. Aczel-Alsina triangular norm (TN) and conorm (TCN) are novel and proved to be more efficient than other existing TNs and TCNs. In our article, we establish the concept of a T-spherical fuzzy Aczel-Alsina graph (TSFAAG). We described the energy of TSFAAG along with the splitting and shadow energy of TSFAAG. Furthermore, we figured out the Randić energy of TSFAAG and obtained some useful results. Moreover, we give the notion of the Aczel-Alsina digraph (TSFAADG). To see the significance of the proposed TSFAADGs, we employed the energy and Randić energy of TSFAADGs for solving the problem of selecting the best investing company by using a decision-making algorithm. The sensitivity analysis of the variable parameters is also discussed and where the effect on ranking results is studied. To see the effectiveness of the proposed work, we did a comparative study and established some remarks.

On polynomial symmetry algebras underlying superintegrable systems in Darboux spaces

We review three different approaches to polynomial symmetry algebras underlying superintegrable systems in Darboux spaces. The first method consists of using deformed oscillator algebra to obtain finite-dimensional representations of quadratic algebras. This allow one to gain information on the spectrum of the superintegrable systems. The second method has similarities with the induced module construction approach in the context of Lie algebras and can be used to construct infinite dimensional representations of the symmetry algebras. Explicit construction of these representations is a non-trivial task due to the non-linearity of the polynomial algebras. This method allows the construction of states of the superintegrable systems beyond the reach of separation of variables. As a result, we are able to construct a large number of states in terms of Airy, Bessel and Whittaker functions which would be difficult to obtain in other ways. We also discuss the third approach which is based on the notion of commutants of subalgebras in the enveloping algebra of a Poisson algebra or a Lie algebra. This allows us to discover new superintegrable models in the Darboux spaces and to construct their integrals and symmetry algebras via polynomials in the enveloping algebras.

Algebraic Reduction of Hidden Markov Models

The problem of reducing a Hidden Markov Model (HMM) to one of smaller dimension that exactly reproduces the same marginals is tackled by using a system-theoretic approach. Realization theory tools are extended to HMMs by leveraging suitable algebraic representations of probability spaces. We propose two algorithms that return coarse-grained equivalent HMMs obtained by stochastic projection operators: the first returns models that exactly reproduce the single-time distribution of a given output process, while in the second the full (multi-time) distribution is preserved. The reduction method exploits not only the structure of the observed output, but also its initial condition, whenever the latter is known or belongs to a given subclass. Optimal algorithms are derived for a class of HMM, namely observable ones.

A sedenion algebraic representation of three colored fermion generations

Three generations of fermions with SU(3) C symmetry are represented algebraically in terms of the algebra of sedenions, 𝕊, generated from the octonions, 𝕆, via the Cayley-Dickson process. Despite significant recent progress in generating the Standard Model gauge groups and particle multiplets from the four normed division algebras, an algebraic motivation for the existence of exactly three generations has been difficult to substantiate. In the sedenion model, one generation of leptons and quarks with SU(3) C symmetry is represented in terms of two minimal left ideals of ℂℓ(6), generated from a subset of all left actions of the complex sedenions on themselves. Subsequently, the finite group S 3, which are automorphisms of 𝕊 but not of 𝕆, is used to generate two additional generations. The present paper highlight the key aspects and ideas underlying this construction.

On the integrality of locally algebraic representations of GL2(D)

Emerton's theory of Jacquet modules for locally analytic representations provides necessary conditions for the existence of integral structures in locally analytic representations. These conditions are also expected to be sufficient for the integrality of generic irreducible locally algebraic representations. In this article, we prove the sufficiency of Emerton's conditions for some tamely ramified locally algebraic representations of GL2(D) where D is a p-adic division algebra.

Derivations and Representations of Commutative Algebras Verifying a Polynomial Identity of Degree Five

In this paper we study a class of commutative non associative algebras satisfying a polynomial identity of degree five. We show that under the assumption of the existence of a non-zero idempotent, any commutative algebra verifying such an identity admits a Peirce decomposition. Using this decomposition we proceeded to the study of the derivations and representations of algebras of this class.

Investigation of Seventh Grade Students’ Transition Skills among Representations

This descriptive case study was aimed at revealing the level of students’ transition skills among representations (verbal, numeric, algebraic, graphic) at the completion of 7th grade. The participants included 133 students attending 7th grade at a state school located in Ankara in Turkey. Students’ skills in transitioning between representations were tested using the “Test of Skills in Transitioning between Representations (TSTBR)”, which included 12 open-ended questions and analyzed student answers through a graded rubric developed by the researcher. Descriptive statistics such as frequency and percentage were used to examine students’ skills in making transitions between representations. Thus, it was determined in the findings that a majority of students had moderate transition skills. According to the mean scores, the number of students with low representation transition skills was greater than the number of students with high representation transition skills. Also, the students were most successful at transitioning from other representations to algebraic, verbal, numeric, and graphic representations, respectively. The mean scores for each representation transition were examined and possible reasons for the difficulties that students faced were discussed.

Deformations of Pairs of Kleinian Singularities

Abstract Kleinian singularities, that is, the varieties corresponding to the algebras of invariants of Kleinian groups are of fundamental importance for Algebraic geometry, Representation theory, and Singularity theory. The filtered deformations of these algebras of invariants were classified by Brieskorn (the commutative case) and Losev (the general case). To an inclusion of Kleinian groups, there is the corresponding inclusion of algebras of invariants. We classify deformations of these inclusions when the smaller subgroup is normal in the larger.

On the Asymptotics of Multiplicities for Large Tensor Product of Representations of Simple Lie Algebras

On the Asymptotics of Multiplicities for Large Tensor Product of Representations of Simple Lie Algebras

McKay correspondence, cohomological Hall algebras and categorification

Let π : Y → X \pi \colon Y\to X denote the canonical resolution of the two dimensional Kleinian singularity X X of type ADE. In the present paper, we establish isomorphisms between the cohomological and K-theoretical Hall algebras of ω \omega -semistable properly supported sheaves on Y Y with fixed slope μ \mu and ζ \zeta -semistable finite-dimensional representations of the preprojective algebra of affine type ADE of slope zero respectively, under some conditions on ζ \zeta depending on the polarization ω \omega and μ \mu . These isomorphisms are induced by the derived McKay correspondence. In addition, they are interpreted as decategorified versions of a monoidal equivalence between the corresponding categorified Hall algebras. In the type A case, we provide a finer description of the cohomological, K-theoretical and categorified Hall algebra of ω \omega -semistable properly supported sheaves on Y Y with fixed slope μ \mu : for example, in the cohomological case, the algebra can be given in terms of Yangians of finite type ADE Dynkin diagrams.

Almost representations of algebras and quantization

abstract: We introduce the notion of almost representations of Lie algebras and quantum tori, and establish an Ulam-stability type phenomenon: every irreducible almost representation is close to a genuine irreducible representation. As an application, we prove that geometric quantizations of the two-dimensional sphere and the two-dimensional torus are conjugate in the semi-classical limit up to a small error.

Middle school students' development of an understanding of the concept of function using an applet with no algebraic representations

AbstractMiddle school students (n = 144) worked with an applet specially designed to introduce the concept of function without using algebraic representations. The purpose of the study was to examine whether the applet would help students understand function as a relationship between a set of inputs and a set of outputs and to begin to develop a definition of function based on that relationship. Results indicate that, by focusing on consistency of the outputs, the students, at a rate of approximately 80%, are able to distinguish functions from nonfunctions. Also, students showed some promise in recognizing constant functions as functions, a known area of common misconceptions. Students' main conceptual difficulty, likely caused by the context, was accepting nonintuitive outputs even if those outputs were consistent.

Representations of superassociative algebras by commutative functions with different types

For fixed integers [Formula: see text] and [Formula: see text], we study the faithful representation of superassociative algebras by two kinds of functions of the arity [Formula: see text], the so called [Formula: see text]-commutative and [Formula: see text]-commutative. The automorphism on the algebra of all [Formula: see text]-commutative functions is determined. In general, an isomorphism from a superassociative system which consists of a family of nonempty sets and a family of operations satisfying the axiom of superassociativity to a system of full multiplace functions is discussed. Particularly, we also provide necessary and sufficient conditions under which a superassociative system and a system of [Formula: see text]-commutative functions are isomorphic. With these results, a close connection with the theory of terms and tree languages is described.

Тензорные расширения уравнений Лакса

The Lax equations dL/dt = [M,L] play an important role in
 the integrability theory of nonlinear evolution equations and
 quantum dynamics. In this work, tensor extensions of the
 Lax equations are suggested with M : V → V and L :
 Tk(V ) → V , k = 1, 2, . . ., on a complex vector space V .
 These extensions belong to the generalised class of Lax equations
 (introduced earlier by Bordemann) dL/dt = ρk(M)L
 where ρk is a representation of a Lie algebra. The case k = 1,
 ρ1 = ad corresponds to the usual Lax equations. The extended
 Lax pairs are studied from the point of view of isomorphic
 deformations of multilinear structures, conservation
 laws, exterior algebras and cochain symmetries.

Decidability of flow equivalence and isomorphism problems for graph C*-algebras and quiver representations

We note that the deep results of Grunewald and Segal on algorithmic problems for arithmetic groups imply the decidability of several matrix equivalence problems involving poset-blocked matrices over Z (the integers). Consequently, results of Eilers, Restorff, Ruiz and Sørensen imply that isomorphism and stable isomorphism of unital graph C*-algebras (including the Cuntz-Krieger algebras) are decidable. One can also decide flow equivalence for shifts of finite type and isomorphism of Z-quiver representations (i.e., finite diagrams of homomorphisms of finitely generated abelian groups).

First-level hypergame for investigating two decision-maker conflicts with unknown misperceptions of preferences within the framework of GMCR

Misunderstandings of preferences are ubiquitous in various types of conflicts, and such misperceptions often involve uncertainty due to the limited cognitive abilities and information asymmetry of decision-makers. This research proposes a novel hypergame analysis approach based on the graph model for conflict resolution to address challenging disputes involving unknown misunderstandings of preferences. Specifically, a first-level hypergame model with two decision-makers is intentionally developed in this study to handle the uncertainty of preference misunderstandings in conflicts. The logical definitions of stabilities in the first-level hypergame are put forward to portray various decision-making behaviors of stakeholders when facing the vagueness of preference misperceptions, and efficiently determine possible resolutions. Algebraic representations of stabilities are also designed for the first time to enhance the computational efficiency of hypergame analyses and facilitate the development of decision support systems. Furthermore, more general definitions of individual and overall equilibria and their algebraic representations are purposefully proposed within the first-level hypergame model. This allows the decision-makers involved in a conflict to hold stable under different solution concepts. Finally, this novel approach is applied to an environmental dispute to demonstrate its practicality in addressing real-world conflicts with uncertain misperceptions of preferences. The case study shows that the uncertainty of preference misunderstandings can affect the optimal strategy selections of decision-makers and result in changes to the equilibria or outcomes of conflicts.