Commutativity rules play an essential role when building multirate signal processing systems. In this letter, we focus on the interchangeability of block decimators and expanders. We, formally, prove that commutativity between these two operators is possible if and only if the data blocks are of an equal length corresponding to the greatest common divisor of the integer decimation and expansion...

Let R be a ring with center Z(R). We write the commutator [x, y] = xy− yx, (x, y ∈ R). The following commutator identities hold: [xy,z] = x[y,z] + [x,z]y; [x, yz] = y[x,z] + [x, y]z for all x, y,z ∈ R. We recall that R is prime if aRb = (0) implies that a= 0 or b = 0; it is semiprime if aRa = (0) implies that a = 0. A prime ring is clearly a semiprime ring. A mapping f : R→ R is called centrali...

It is shown that coherence conditions for monoidal categories concerning associativity are analogous to coherence conditions for strict symmetric monoidal categories, where associativity arrows are identities. Mac Lane’s pentagonal coherence condition for associativity is decomposed into conditions concerning commutativity, among which we have a condition analogous to naturality and a degenerat...

In this paper, we prove common fixed point theorems for hybrid pairs of mappings satisfying an implicit relation in 2-metric spaces by using a new commutativity condition i.e. weak commutativity of type (KB). We also prove common fixed point theorem, of Gregus type for hybrid pairs of maps by using weak commutativity of type (KB) in 2-metric spaces. We extend, improve and generalize many known ...

Commutativity of operations is the better way to provide a high degree of concurrency on shared data types. In this short paper, we present a technique to increase concurrency using operational transformations. This technique enforces commutativity even though the operations do not naturally commute. We report our experience on (i) automatically verifying the correctness of this transformation-...

In an invited paper, Baksalary [Algebraic characterizations and statistical implications of the commutativity of orthogonal projectors. In: T. Pukkila, S. Puntanen (Eds.), Proceedings of the Second International Tampere Conference in Statistics, University of Tampere, Tampere, Finland, [2], pp. 113–142] presented 45 necessary and sufficient conditions for the commutativity of a pair of orthogon...

A simple proof of associativity and commutativity of LR-coefficients (or the hive ring) In this paper we propose a simple bijective proof of associativity and commutativity of Littlewood-Richardson coefficient or the hive ring ([13]).

In the Map-Reduce programming model for data parallel computation in a cloud environment, the reducer phase is responsible for computing an output key-value pair, given a sequence of input values associated with a particular key. Because of non-determinism in transmitting key-value pairs over the network, the inputs may not arrive at a reducer in a fixed order. This gives rise to the reducer co...

then we term the elements x and y n-commutative. It is not difficult to verify that »-commutativity and (1—»)-commutativity are equivalent properties of the elements x and y, that ( — ̂ -commutativity implies ordinary commutativity, and that commuting elements are »-commutative. From any concept and property involving the fact that certain elements [or functions of elements] commute, one may der...

How can we enhance the understanding of abstract mathematical principles in elementary school? Different studies found out that nonsymbolic estimation could foster subsequent exact number processing and simple arithmetic. Taking the commutativity principle as a test case, we investigated if the approximate calculation of symbolic commutative quantities can also alter the access to procedural an...