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Legendre's three-square theorem states which numbers can be expressed as the sum of three squares; Jacobi's four-square theorem gives the number of ways that a number can be represented as the sum of four squares. For the number of representations of a positive integer as a sum of squares of k integers, see Sum of squares function.
A snippet of JavaScript code with keywords highlighted in different colors The syntax of JavaScript is the set of rules that define a correctly structured JavaScript program. The examples below make use of the log function of the console object present in most browsers for standard text output .
The number of ways to write a natural number as sum of two squares is given by r 2 (n).It is given explicitly by = (() ())where d 1 (n) is the number of divisors of n which are congruent to 1 modulo 4 and d 3 (n) is the number of divisors of n which are congruent to 3 modulo 4.
For example, the function PT tests whether each row of ⍵ is a Pythagorean triplet (by testing whether the sum of squares equals twice the square of the maximum). PT ← { ( + / ⍵ * 2 ) = 2 × ( ⌈ / ⍵ ) * 2 } PT 3 4 5 1 x 4 5 3 3 11 6 5 13 12 17 16 8 11 12 4 17 15 8 PT x 1 0 1 0 0 1
The explained sum of squares (ESS) is the sum of the squares of the deviations of the predicted values from the mean value of a response variable, in a standard regression model — for example, y i = a + b 1 x 1i + b 2 x 2i + ... + ε i, where y i is the i th observation of the response variable, x ji is the i th observation of the j th ...
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The general regression model with n observations and k explanators, the first of which is a constant unit vector whose coefficient is the regression intercept, is = + where y is an n × 1 vector of dependent variable observations, each column of the n × k matrix X is a vector of observations on one of the k explanators, is a k × 1 vector of true coefficients, and e is an n× 1 vector of the ...
A sum-of-squares optimization program is an optimization problem with a linear cost function and a particular type of constraint on the decision variables. These constraints are of the form that when the decision variables are used as coefficients in certain polynomials , those polynomials should have the polynomial SOS property.