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The smallest friendly number is 6, forming for example, the friendly pair 6 and 28 with abundancy σ(6) / 6 = (1+2+3+6) / 6 = 2, the same as σ(28) / 28 = (1+2+4+7+14+28) / 28 = 2. The shared value 2 is an integer in this case but not in many other cases. Numbers with abundancy 2 are also known as perfect numbers. There are several unsolved ...
Hasse diagram of the natural numbers, partially ordered by "x≤y if x divides y".The numbers 4 and 6 are incomparable, since neither divides the other. In mathematics, two elements x and y of a set P are said to be comparable with respect to a binary relation ≤ if at least one of x ≤ y or y ≤ x is true.
To prove that this condition is sufficient to guarantee existence of a compatible second-order tensor field, we start with the assumption that a field exists such that =. We will integrate this field to find the vector field v {\displaystyle \mathbf {v} } along a line between points A {\displaystyle A} and B {\displaystyle B} (see Figure 2), i.e.,
For instance, the family of functions from the natural numbers to the integers is the uncountable set of integer sequences. The subfamily { f ∈ Z N : f has finite support } {\displaystyle \left\{f\in \mathbb {Z} ^{\mathbb {N} }:f{\text{ has finite support }}\right\}} is the countable set of all integer sequences that have only finitely many ...
Consider -dimensional Euclidean space with its usual Borel topology and σ-algebra. Consider a collection of Gaussian measures Γ = { γ i ∣ i ∈ I } , {\displaystyle \Gamma =\{\gamma _{i}\mid i\in I\},} where the measure γ i {\displaystyle \gamma _{i}} has expected value ( mean ) m i ∈ R n {\displaystyle m_{i}\in \mathbb {R} ^{n}} and ...
The general notion of a congruence is particularly useful in universal algebra. An equivalent formulation in this context is the following: [4] A congruence relation on an algebra A is a subset of the direct product A × A that is both an equivalence relation on A and a subalgebra of A × A. The kernel of a homomorphism is always a congruence ...
definition remarks rational equivalence Z ~ rat Z' if there is a cycle V on X × P 1 flat over P 1, such that [V ∩ X × {0}] − [V ∩ X × {∞}] = [Z] − [Z' ]. the finest adequate equivalence relation (Lemma 3.2.2.1 in Yves André's book [2]) "∩" denotes intersection in the cycle-theoretic sense (i.e. with multiplicities) and [.] denotes the cycle associated to a subscheme. see also ...
In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues. On the other hand, in numerical algorithms for differential equations the concern is the growth of round-off errors and/or small fluctuations in initial data which might ...