Additional Commutation Research Articles

Young Children's Understanding of Addition Concepts

Children's knowledge of concrete versions of additive composition, commutativity and associativity was investigated in two studies. In Study 1, 24 four- to five-year-olds and 25 five- to six-year-olds judged the equivalence of conceptually related addition problems presented using groups of objects. In Study 2, 45 five- to six-year-olds judged related problems and solved addition problems. Both studies indicated that concrete versions of principles were salient to most children although associativity was more difficult than commutativity and there were considerable individual differences in children's understanding. Study 1 results indicated that schoolchildren were more accurate at recognising additive composition than preschoolers and Study 2 results suggested that commutativity knowledge was related to using advanced counting strategies for solving addition problems. Overall, the research supports the claim that examining early knowledge of addition principles provides important insights into children's emerging part-whole knowledge and mathematical development.

Notes on Whitehead space of an algebra

Let R be a ring, and denote by [R, R] the group generated additively by the additive commutators of R. When Rn = Mn(R) (the ring of n × n matrices over R), it is shown that [Rn, Rn] is the kernel of the regular trace function modulo [R, R]. Then considering R as a simple left Artinian F‐central algebra which is algebraic over F with Char F = 0, it is shown that R can decompose over [R, R], as R = Fx + [R, R], for a fixed element x ∈ R. The space R/[R, R] over F is known as the Whitehead space of R. When R is a semisimple central F‐algebra, the dimension of its Whitehead space reveals the number of simple components of R. More precisely, we show that when R is algebraic over F and Char F = 0, then the number of simple components of R is greater than or equal to dimF R/[R, R], and when R is finite dimensional over F or is locally finite over F in the case of Char F = 0, then the number of simple components of R is equal to dimF R/[R, R].

$ \sigma $-derivations on prime near-rings

The literatrue on near-rings contains a number of theorems asserting that certain conditions implying commutativity in rings imply multiplicative or additive commutativity in special classes of near-rings. H. E. Bell and G. Mason in [2] added to this body of results several commutativity theorems for near-rings admitting suitably-constrained derivations. In this paper we generalize some of their results to a subclass of prime near-rings admitting suitably-constrained $ \sigma $-derivations, where $ \sigma $ is an automorphibm of the prime near-ring.

Kindergartners' Understanding of Additive Commutativity within the Context of Word Problems

Baroody and Gannon (1984) proposed that children's understanding of additive commutativity progresses through several levels of understanding based on a unary view of addition (change meaning) before developing a "true" level of understanding based on a binary conception (part-whole meaning). Resnick (1992) implied that children have both a unary and a binary conception of additive commutativity from the earliest stages of development. Fifty-three 5- and 6-year-old (M = 6-0) kindergartners' unary and binary understanding of additive commutativity was investigated using performance on tasks involving change-add-to and part-part-whole word problems, respectively. The data were inconsistent with the predictions of both models and suggest three alternate theoretical explanations. Moreover, the data indicate that success on a task involving change-add-to problems may be a more rigorous test of understanding of additive commutativity than that involving part-part-whole problems.

A robust multisyringe system for process flow analysis. Part II. A multi-commuted injection system applied to the photometric determination of free acidity and iron(III) in metallurgical solutions.

A new software-controlled volume-based system for sample introduction in process flow injection analysis was developed. By using a multi-syringe burette coupled with one or two additional commutation valves, the multi-commuted injection of precise sample volumes was accomplished. Characteristics and performance of the injection system were studied by injecting an indicator in a buffered carrier. Three configurations were implemented in order to achieve two different tasks: the single injection of a sample in a two- or three-channels manifold, and the dual injection into different streams. The two channel flow system using the single injection was applied to the determination of free acidity in diluted samples containing high levels of iron(III), by employing the single point titration methodology. The precipitation of ferric hydroxide was prevented using the ammonium and sodium salts of oxalate and acetate as buffer titrant. Methyl Red was employed as indicator. The procedure allows determination of acid concentration in solutions with a Fe(III)/H+ molar ratio up to 0.2. Samples with higher Fe(III)/H+ molar ratios were spiked with a known strong acid at dilution. The three-channel configuration was applied to the determination of ferric ions, using, as reagent, a merging mixture of sulfuric acid and potassium thiocyanate. The double injection system was implemented in series in a single (three-channel) manifold in such a way that a different injection volume and a changed reagent were used for each analyte. It was applied to the separated or sequential determination of free acidity and ferric ions. In this configuration, iron(III) was determined using 0.5-0.7% (w/v) sodium salicylate solution as reagent. The systems can operate at up to 100, 84 and 78 injections per hour, respectively. Determinations on synthetic and process samples compared well with the reference values and procedures. Recoveries of 95-102% with a maximum RSD value of 5.4% were found for acidity. The respective values obtained for iron determinations were 96-105% and 4.3%.

On Additive Maps and Commutativity in Rings

We study additive maps which are skew-commuting or skew-centralizing on appropriate subsets of a ring R; and we investigate commutativity in prime and semiprime rings admitting a nonzero derivation d such that [d(x),d(y)] = 0 for all x,y in some nonzero one-sided ideal. This paper has two main parts. The first, motivated by a recent result of Brešar [3] on triviality of skew-commuting additive maps on prime rings, is a study of additive maps which are skew-commuting or skew-centralizing on subsets of certain rings. The second continues a study, begun years ago by Herstein [7], of prime and semiprime rings R admitting a nonzero derivation d such that d(x)d(y) − d(y)d(x) = 0 for all x, y in a suitably chosen subset of R.

The role of conceptual understanding in children's addition problem solving.

The study examined the relationship between children's conceptual understanding and addition problem-solving procedures. Forty-eight 6- to 8-year-olds solved addition problems and, in a 2nd task, were prompted to judge whether a puppet could use the arithmetic properties of one problem to solve the next problem. Relational properties between consecutive problems were manipulated to reflect aspects of additive composition, commutativity, and associativity principles. Conceptual understanding was assessed by the ability to spontaneously use such relational properties in problem solving (Task 1) and to recognize and explain them when prompted (Task 2). Results revealed that conceptual understanding was related to using order-indifferent, decomposition, and retrieval strategies and speed and accuracy in solving unrelated problems. The importance of conceptual understanding for addition development is discussed.

On Additive Commutator Groups in Division Rings

Let D be a division ring with center F and denote by [D,D] the group generated additively by additive commutators. First, it is shown that in zero characteristic, D is algebraic over F if and only if each element of [D,D] is algebraic over F. We conjecture that this assertion is true for any characteristic. Also, as a generalization of Jacobson’s Theorem it is proved that D is an F-central division ring if and only if all its additive commutators are of bounded degree over F. Furthermore, we study the F- vector space D/[D,D] and show that dimFD/[D,D] ≤ 1 if D is algebraic over F and char F = 0. We then prove that any algebraic division ring contains a separable additive commutator over F except in one special case. Finally, the existence of primitive elements in [D, D] is studied for finite separable extensions of F in D.

The role of conceptual understanding in children's addition problem solving.

The study examined the relationship between children's conceptual understanding and addition problem-solving procedures. Forty-eight 6- to 8-year-olds solved addition problems and, in a 2nd task, were prompted to judge whether a puppet could use the arithmetic properties of one problem to solve the next problem. Relational properties between consecutive problems were manipulated to reflect aspects of additive composition, commutativity, and associativity principles. Conceptual understanding was assessed by the ability to spontaneously use such relational properties in problem solving (Task 1) and to recognize and explain them when prompted (Task 2). Results revealed that conceptual understanding was related to using order-indifferent, decomposition, and retrieval strategies and speed and accuracy in solving unrelated problems. The importance of conceptual understanding for addition development is discussed.

Unitarily-invariant linear spaces in C*-algebras

Characterisations and containment results are given for linear subspaces of a unital C*-algebra that are invariant under conjugation by sets of unitary elements of the algebra. The (unitarily-invariant) linear span of the projections in a simple, unital C*-algebra having non-scalar projections is shown to contain all additive commutators of the algebra and, in certain cases, to be equal to the algebra.

On Optimizing a Class of Multi-Dimensional Loops with Reduction for Parallel Execution

This paper addresses the compile-time optimization of a form of nested-loop computation that is motivated by a computational physics application. The computations involve multi-dimensional surface and volume integrals where the integrand is a product of a number of array terms. Besides the issue of optimal distribution of the arrays among the processors, there is also scope for reordering of the operations using the commutativity and associativity properties of addition and multiplication, and the application of the distributive law to significantly reduce the number of operations executed. A formalization of the operation minimization problem and proof of its NP-completeness is provided. A pruning search strategy for determination of an optimal form is developed. An analysis of the communication requirements and a polynomial-time algorithm for determination of optimal distribution of the arrays are also provided.

On Wick algebras with additional twisted commutation relations

We study the construction of the C-symmetric creation and annihilation operators, which form a Wick algebra with some additional relations. This Wick algebra is an algebra of the operators which act on the associative algebra of polynomials with noncommutative variables divided by an ideal generated by quadratic relations between the variables . As a result the consistency conditions for such a representation are obtained. The creators and annihilator have to preserve ideal , where , which leads us to consistency conditions.

Products of matrices with prescribed nullities and traces

First necessary and sufficient conditions are presented for a square matrix over an arbitrary field to be a product of k ≥ 2 matrices with prescribed traces. Then the same is done for products in which both the nullities and traces of the factors are prescribed. Finally these results are applied to products of additive commutators to obtain similar results.

ALGEBRAIC PROCEDURES USED BY 13-TO-15-YEAR-OLDS

Students of Grade 7 were given a test followed by individual interviews; at the end of Grade 8 the same students were subject to an analogous test and interviews. Each student had to simplify certain algebraic expressions. This article is focussed on what types of procedures were used by the students performing the task and how they were explained during the interviews. The author identifies seven types of procedures used by students, labelled: (A) Automatization, (F) Formulas, (GS) Guessing-Substituting, (PM) Preparatory Modification of the expression (this includes a subtype: Atomization), (C) Concretization, (R) Rules, (QR) Quasi-rules. Part of the students' procedures led to correct results, others were wrong. Most of the procedures appeared spontaneous in the sense that they had not been taught in the classroom. Prior to the tests, the teachers (in accordance with the curriculum) had done their best to explain the validity of algebraic transformations by referring to the commutativity of addition and multiplication, distributivity, and to geometric interpretation; however, the interviewees (even explicitly asked) seldom used such arguments.

Conjugacy and factorization results on matrix groups

In this survey paper, we present (mainly without proof) a number of results on conjugacy and factorization in general linear groups over fields and commutative rings. We also present the additive analogue in matrix rings of some of these results. The first section deals with the question of expressing elements in the commutator subgroup of the general linear group over a field as (simple) commutators. In Section 2, the same kind of problem is discussed for the general linear group over a commutative ring. In Section 3, the analogous question for additive commutators is discussed. The case of integer matrices is given special emphasis as this is an area of current interest. In Section 4, factorizations of an element A ∈ GL(n, F ) (F a field) in which at least one of the factors preserves some form (e.g. is symmetric or skew-symmetric) is considered. An application to the size of abelian subgroups of finite p-groups is presented. In Section 5, a curious interplay between additive and multiplicative commutators in Mn(F ) (F a field) is identified for matrices of small size and a general factorization theorem for a polynomial using conjugates of its companion matrix is presented.

Inverter motor with sinusoidal starting currents ‐ a survey

AbstractInverter motors need only simple line‐ and machine‐commutated converters and have proved good performance for drives in the MW range. They cause high torque ripple, which is disturbing only at low frequencies, and require additional commutation devices under heavy start‐up conditions. The also very economic cyclo‐converters are only useful for machine frequencies up to about 40 % of line frequency. The new systems proposed here combine the advantages of both drives with two DC links only. The same circuits operate as a cy‐cloconverter at start‐up and low frequencies of the motor, and as a current‐source inverter at high speeds. The switchover possibilities from one operation mode to the other without any transient reaction are shown. The necessary expenditure compared to the conventional inverter motor is presented.

Permutation identity near-rings and ?localized? distributivity conditions

A near-ringR is said to satisfy apermutation identity if there is some non-identity permutation σ of lengthn such that Πaj=Πaσ(j), for eacha1, ...,an∈R. Numerous examples of permutation identity near-rings are given. The theory is then developed making use of various “localized” distributive conditions, which include as special cases most of the standard global ones (e. g., d. g., pseudo-distributive). These localized conditions only assume distributivity among the elements of certain special (and often “small”) sets. Particularly useful for such sets are powers of the ideals generated by the sets of Lie commutators, additive commutators, or distributive elements. Examples are given where a localized condition holds yet none of the usual global ones do. Results are obtained concerning prime, semiprime, or maximal ideals as well as regular, simple, or subdirectly irreducible near-rings.

Special Additive and Multiplicative Commutator Representations for Diagonal Operators

Let 𝒟 be a dense linear subspace of a separable Hilbert space and let ℒ︁+(𝒟) be the maximal Op*-Algebra on 𝒟. The paper deals with a class of diagonal operators of ℒ︁+(𝒟), for which we can prove some results concerning special commutator representations. For this end ideas of [1] are generalized.

Additive and Multiplicative Commutators in Algebras of Unbounded Operators

Additive and Multiplicative Commutators in Algebras of Unbounded Operators

Medial near-rings

In this paper we discuss (left) near-rings satisfying the identities:abcd=acbd,abc=bac, orabc=acb, called medial, left permutable, right permutable near-rings, respectively. The structure of these near-rings is investigated in terms of the additive and Lie commutators and the set of nilpotent elementsN (R). For right permutable and d.g. medial near-rings we obtain a “Binomial Theorem,” show thatN (R) is an ideal, and characterize the simple and subdirectly irreducible near-rings. “Natural” examples from analysis and geometry are produced via a general construction method.