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Development of methods for calculating non-stationary heat transfer between multilayer structures
Published in Vladimir Litvinenko, Advances in Raw Material Industries for Sustainable Development Goals, 2020
A. Gorshkov, A. Demidov, A. Makarov
The associativity of the ring R⊗ follows from the associativity of the ring R⊗‾. Multiplication in the sense of (3) is associated with addition due to the laws of distributivity. R⊗ - a ring with a unit element, while the unit element of the ring R⊗ is the function Et,S≡1.
General linear codes
Published in Jürgen Bierbrauer, Introduction to Coding Theory, 2016
Why is Theorem 3.2 true? As addition and multiplication in Fp are inherited from the integers, it is clear that distributivity holds and that addition and multiplication are commutative. The only problem is the existence of the inverse (equivalently the absence of 0-divisors). A number is ≠ 0 mod p if it is not divisible by p. The nonexistence of 0-divisors is therefore equivalent to the following statement: if integers a and b both are not divisible by p, then a · b is not divisible by p. This is clearly true. It is one way of expressing the defining property of a prime, and it certainly is not true for composite numbers (see the example for n = 4 above).
Linear Algebra
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
The properties three-dimensional vectors are generalized to spaces of more than three dimensions in linear algebra courses. The properties roughly outlined above need to be preserved. So, we will start with a space of vectors and the operations of addition and scalar multiplication. We will need a set of scalars, which generally come from some field. However, in our applications, the field will either be the set of real numbers or the set of complex numbers. A field is a set together with two operations, usually addition and multiplication, such that we have Closure under addition and multiplicationAssociativity of addition and multiplicationCommutativity of addition and multiplicationAdditive and multiplicative identityAdditive and multiplicative inversesDistributivity of multiplication over addition
A parameter method for linear algebra and optimization with uncertainties
Published in Optimization, 2020
Nam Van Tran, Imme van den Berg
To be concrete, with respect to addition and multiplication external numbers form completely regular semigroups [21,22] (i.e. elements have well-behaved individual neutral elements) and distributivity does not fully hold but can be completely characterized [23]. Order is total and respected by the operations, with a form of Dedekind-completeness; this implies there are well-defined bounds to the imprecision of external convex sets in terms of external numbers [1]. Especially relevant for problems with more variables is that in higher dimensional space neutrices come in the form of modules which also have a dimension [24,25]. Comparing with other approaches, the calculus of external numbers is more efficient than ordinary asymptotics and the Art of Neglecting of Van der Corput, which for instance also lacks total order, due to the fact that his neutrices are sets of functions and not of numbers. With respect to random methods, we note that the functional operations on probability distributions are usually complex. Calculus for fuzzy numbers does not treat all representatives equally and depends on a possibly not so effective calculus of membership functions. Multiparametric methods are again functional with all its operational complications. The set-theoretic calculus of intervals behaves better on operations, but has problems of subdistributivity and intersection, and frictions between algebra and order.
Stability and stabilisation of networked pairing problem via event-triggered control
Published in International Journal of Control, 2022
If n = p, the STP defined in Definition 2.1 degenerates to the conventional matrix product. Hence, STP is a generalisation of the conventional matrix product. Hence, we adopt as a convention that .Most of the major properties of conventional matrix product, such as associativity and distributivity etc., remain available for STP.
Emergent modelling to introduce the distributivity property of multiplication: a design research study in a primary school
Published in International Journal of Mathematical Education in Science and Technology, 2022
The distributivity property of multiplication over addition (DP) is defined as with for simplicity whole numbers. DP is basilar in algebra, for the study of polynomials and vector spaces, and for the definition of the structure of field, underpinning flexible mental calculations and algebraic understanding (Ding & Li, 2010; Carpenter et al., 2005; Young-Loveridge, 2005; Lampert, 1986). Moreover, it characterizes operations between whole numbers, and so it might be faced since the first introduction of multiplication, also in primary school (Maffia & Mariotti, 2020). Despite the importance of DP, it still remains hard for students to learn it (Squire et al., 2004). One of the reasons for such difficulty in students’ learning of DP can be attributed to its abstractness and lack of close relevance to learners’ lives (Ding & Li, 2014). Instead, in order to build a mental image of DP, concretization by contextual and visual representations should help students to structure and organize their reasoning (Yackel, 2001). Examples of these representations had been introduced through different multiplicative situations, such as equal groups and rectangular arrays (Izsak, 2004; Greer, 1992). Squire et al. (2004) suggested that the representation of equal groups might be a natural way for introducing DP to young students. This result was confirmed also by Schifter et al. (2008) and Larsson (2015). However, even if the equal groups representation appears to be more intuitive for young students than the rectangular array model, other researchers have shown that the rectangular array model can support students’ understanding of DP increasing their calculation skills (Maffia & Mariotti, 2020; Barmby et al., 2009; Izsak, 2004; Freudenthal, 1973).