linear function

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linear function

[′lin·ē·ər ′fəŋk·shən]
(mathematics)

Linear Function

 

a function of the form y = kx + b. The basic property of a linear function is that an increment in the function is proportional to the corresponding increment in the independent variable. The graph of a linear function is a straight line. If we use the same units on the coordinate axes, then the coefficient k (the slope) is equal to the tangent of the angle formed by the line with the Ox axis (k = tan ϕ; see Figure 1) and b is the segment cut off by the line on the Oy axis. When b = 0, the linear function is called homogeneous; its graph depicts the direct proportion: y = kx.

Figure 1

Linear functions are widely used in physics and engineering in order to represent (often approximately) the dependence of various quantities. Linear functions of several variables are also studied. Homogeneous linear functions of several variables are called linear forms. Homogeneous linear functions which send vectors to vectors are called linear transformations.

linear function

A recursive function is linear if it is of the form

f x = if p x then q x else h f x

where h is a "linear functional" which means that

(1) for all functions, a, b c and some function ht

h (if a then b else c) = if ht a then h b else h c

Function ht is known as the "predicate transformer" of h.

(2) If for some x,

h (\ y . bottom) x /= bottom

then

for all g, ht g x = True.

I.e. if h g x terminates despite g x not terminating then ht g x doesn't depend on g.

See also linear argument.
References in periodicals archive ?
It shows that the stochastic differential equation models used in derivative pricing can be interpreted as linear factor models that relate sources of risk to returns, illustrating how Ross' Arbitrage Pricing Theory leads to the absence of arbitrage equations that govern the pricing of derivatives.
Both the linear factor and the quadratic factor of the drying time (DT) had no significant (p>0.
In the latter case, a major problem could occur in a motor installation since current imbalance on one phase (whether or not due to an earth fault) is a linear factor of current rise and therefore measurable.
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As the emissions trading directive says: "The Commission shall review the linear factor and submit a proposal, where appropriate, to the European Parliament and to the Council as from 2020, with a view to the adoption of a decision by 2025.
The approximation substitutes a linear factor for the logistic activation function.
Ludvigson, Federal Reserve Bank of New York, "A Cross-Sectional Test of Linear Factor Models with Time-Varying Risk Premiums"
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By far, the most common model for analyzing this type of items is linear factor analysis (FA).
It does not give any numbers at this stage on the percentage of CO2 quotas to attribute to companies in relation to their 2005 emissions but says that the general volume of quotas granted at EU level from 2013 will have to decrease progressively according to a linear factor defined halfway through the 2008-2012 period in comparison with the total volume of quotas granted by the member states in accordance with national allocation plans for quotas (NAPs) backed by the Commission for this period.
However, it says that the general volume of quotas granted at EU level from 2013 will have to decrease progressively according to a linear factor defined halfway through the 2008-2012 period in comparison with the total volume of quotas granted by the member states in accordance with national allocation plans for quotas (NAPs) backed by the Commission for this period.

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