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- Many studies have investigated how multiple stimuli combine to reach threshold. There are broadly speaking two ways this can occur: additive summation (AS) where inputs from the different stimuli add together in a single mechanism, or probability summation (PS) where different stimuli are detected independently by separate mechanisms.
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For the summation of the sequence of consecutive integers from 1 to 100 one could use an addition expression involving an ellipsis to indicate the missing terms: 1 + 2 + 3 + 4 + ... + 99 + 100.
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Apr 23, 2022 · Use summation notation to express the sum of squares. Many statistical formulas involve summing numbers. Fortunately there is a convenient notation for expressing summation. This section covers the basics of this summation notation.
- Additive Function Definition
- Examples of Additive Functions
- Completely (Totally) Additive Function
- Subadditive Function
- Superadditive Function
- Examples of Superadditive Function
- References
One of the simplest types of arithmetical functionsis the additive function, which has the form f(ab) = f(a) + f(b) for all relatively prime (coprime) positive integers a and b. Relatively primemeans that two integers don’t share any common factors except 1. For example, 6 and 5 are relatively prime, as are 30 and 7. This type of function has some ...
If a and b are relatively prime positive integers, then the logarithm log(ab) = log(a) + log(b) is an additive function. It doesn’t matter what base your log is in. For example: 1. log(10) = log(5) + log(2), 2. ln(6) = ln(3) + ln (2). The image below shows the graph of ln(x) between 0 and 10. It is an additive function because for all positive, cop...
If the function f(ab) = f(a) + f(b) holds even when a and b are not relatively prime, f(x) is completely additive (also called totally additive). For every function f that is completely additive, f(1) = 0.
A subadditive functionis formally defined as one where the following inequality holds (Kadakal, 2020): f (x + y) ≤ f(x) + f(y), for x, y ∈ [0, ∞).
A superadditive function can be defined in terms of a subadditive function. A function is superadditive on an interval if and only if –fis subadditive. To put this in notation, a function is superadditive on an interval if the following inequality holds: f (x + y) ≥ f(x) + f(y) Note that the inequality is reversed: some definitions of superadditive...
Every additive functionis subadditive and superadditive. For example, logb (x) is an additive function, because logb(x, y) = logb x + logb y. A star-shaped function, with respect to the origin, is superadditive. Another example is a convex function f with f(0) ≥ 0 (Bruckner, 1962). Note though, that a superadditive function isn’t always convex, eve...
Brucker, A. (1962). Tests for the Superadditivity of Functions. Pacific J. Math. 10 (1960), no. 4, 1155–1162. Kadakal, H. (2020). Hermite-Hadamard type inequalities for subadditive functions. AIMS Mathematics, 5(2): 930–939. Nathanson, M. Additive Number Theory: Inverse Problems and the Geometry of Sumsets (Graduate Texts in Mathematics) (Vol 165)1...
An infinite series converges and has sum S if the sequence of partial sums, \(\{S_n\}\) converges to S. Thus, if \[S=\lim_{n\rightarrow \infty} \{S_n\}\] then the series converges to S. If \(\{S_n\}\) diverges, then the series diverges. Review Convergence and Divergence of Series Video, (Khan Academy).
There are broadly speaking two ways this can occur: additive summation (AS) where inputs from the different stimuli add together in a single mechanism, or probability summation (PS) where different stimuli are detected independently by separate mechanisms.
- Frederick A. A. Kingdom, Alex S. Baldwin, Gunnar Schmidtmann
- 2015
Apply summation notation to calculate statistics. This notation is called summation notation and appears as: In this notation, the is an expression that contains the index and you plug in 1 and then 2 and then 3 all the way to the last number and then add up all of the results.
The countable additivity axiom states that the probability of a union of a finite collection (or countably infinite collection) of disjoint events * is the sum of their individual probabilities. P (A 1, ∪ A 2 ∪ + A 3 ∪ …).
