The mean is the only single number for which the residuals defined this way sum to zero.

According to the theorem, it is possible to expand the power ( x + y )< sup > n </ sup > into a sum involving terms of the form ax < sup > b </ sup > y < sup > c </ sup >, where the exponents b and c are nonnegative integers with, and the coefficient a of each term is a specific positive integer depending on n and b. When an exponent is zero, the corresponding power is usually omitted from the term.

The slope of the surface adjusts to make all three forces sum to zero.

To quote Anthony and Brackett: Moreover, because the element of water does not move, the sum of all three forces must be zero.

To sum to zero, the force of the water must point oppositely to the sum of the centrifugal and gravity forces, which means the surface of the water must adjust so its normal points in this direction.

An example of an NP-complete problem is the subset sum problem: given a finite set of integers is there a non-empty subset which sums to zero?

To give a proof of a " yes " instance, one must specify a non-empty subset which does sum to zero.

" To give a proof of a " no " instance one must specify a non-empty subset which does sum to zero, which is easily verified.

Since the chain is at rest the sum of these forces must be zero.

When waves are added together, their sum is determined by the relative phases as well as the amplitudes of the individual waves so that the summed amplitude of the waves can have any value between zero and the sum of the individual amplitudes.

* Kirchhoff's voltage law: The directed sum of the electrical potential differences around a loop must be zero.

Otherwise the sum would again be zero due to the antisymmetry, thus representing a physically impossible state.

The mercantilist idea that all trade was a zero sum game, in which each side was trying to best the other in a ruthless competition, was integrated into the works of Thomas Hobbes.

Equivalent technical statements are that the sum total magnetic flux through any Gaussian surface is zero, or that the magnetic field is a solenoidal vector field.

An example of a non-multiplicative function is the arithmetic function r < sub > 2 </ sub >( n )-the number of representations of n as a sum of squares of two integers, positive, negative, or zero, where in counting the number of ways, reversal of order is allowed.

A polynomial is either zero or can be written as the sum of a finite number of non-zero terms.

Thus each polynomial is actually equal to the sum of the terms used in its formal expression, if such a term a < sub > i </ sub > X < sup > i </ sup > is interpreted as a polynomial that has zero coefficients at all powers of X other than X < sup > i </ sup >.

In equilibrium, the forces on the moving block must sum to zero.

The first and last terms on the right hand side are diagonal elements and are zero, and the whole sum is equal to zero.

In order to maintain the steady-state propulsion of the system without either accelerating or decelerating the system, the sum of all the external forces on the system, averaged over the cycle, must be zero.

For example, the sum of the empty set is NULL, meaning zero, the average of the empty set is NULL, meaning undefined, and NULL appearing in the result of a LEFT JOIN can mean " no value because there is no matching row in the right-hand operand ".

Maximize the sum of the values of the items in the knapsack so that the sum of the weights must be less than the knapsack's capacity.

Similarly, the opportunity cost of attending university is the lost wages a student could have earned in the workforce, rather than the cost of tuition, books, and other requisite items ( whose sum makes up the total cost of attendance ).

In calculating the arithmetic mean of a sample, for example, the algorithm works by summing all the data values observed in the sample then divides this sum by the number of data items.

For perceivable items, it says that every perceivable item is a sum of items at a smaller level of complexity.

Type ontological reductionism is the idea that every type of item is a sum type of item, and that every perceivable type of item is a sum of types of items at a lower level of complexity.

of the array. divides the sum by the number of items.

It then performs a prefix sum computation ( a second loop, over the range of possible keys ) to determine, for each key, the starting position in the output array of the items having that key.

However, in modern society it has become common place for restaurants to serve dim sum at dinner time, various dim sum items are even sold as take-out for students and office workers on the go.

According to accounts, the following day a person unknown to residents purchased both items for " a large sum of money ", declaring them " of Asiatic origin ".

:: For these preliminary experiments, as you may recollect, a sum not to exceed $ 200 or $ 250 was to be appropriated, and in accordance with this Mr. Lowe has presented me with the inclosed statement of items, which I think are reasonable, since nothing is charged for labor and time of the aeronaut.

If a kitsune offers a payment or reward that includes money or material wealth, part or all of the sum will consist of old paper, leaves, twigs, stones, or similar valueless items under a magical illusion.

This analysis consists of computation of item difficulties and item discrimination indices, the latter index involving computation of correlations between the items and sum of the item scores of the entire test.

* Constant sum scale – a respondent is given a constant sum of money, script, credits, or points and asked to allocate these to various items ( example: If you had 100 Yen to spend on food products, how much would you spend on product A, on product B, on product C, etc .).

These food items are collectively known as " dim sum ", a varied range of small dishes which may constitute or replace breakfast, brunch or afternoon tea.

Increasingly, though, some games allow for ' resurrection ' or ' cloning ' and the opportunity to regain lost items upon death for a certain sum of ingame currency, typically increasing exponentially the more times the player dies.

Thus, the test information function is simply the sum of the information functions of the items on the exam.

The Likert scale is the sum of responses on several Likert items.

" " Mainstream is more than the sum of all cross-currents and sub-currents, it represents the broadest range of shared meanings and assumptions in the most general, functional and stable way Heavy television viewing may override individual differences and perspectives, creating more of an American ( and increasingly global ) " melting pot " of social, cultural and political ideologies Essentially, the more TV a viewer watches, the more likely it becomes that their opinions of various items in the world will start to mirror those the media portrays.

Expensive items may be placed on two plates, with the price being the sum of the prices of the individual plates.

* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.

Therefore no complete system, i. e. including the electromagnetic modes, can have negative temperatures, since there is no highest energy state, so that the sum of the probabilities of the states would diverge for negative temperatures.

After describing the manifestation of the Gospel in the Ogdoad and Hebdomad, he adds that the Basilidians have a long account of the innumerable creations and powers in the several ' stages ' of the upper world ( diastemata ), in which they speak of 365 heavens and say that " their great archon " is Abrasax, because his name contains the number 365, the number of the days in the year ; i. e. the sum of the numbers denoted by the Greek letters in ΑΒΡΑΣΑΞ according to the rules of isopsephy is 365:

This shows in particular that is a natural number for any natural numbers n and k. There are many other combinatorial interpretations of binomial coefficients ( counting problems for which the answer is given by a binomial coefficient expression ), for instance the number of words formed of n bits ( digits 0 or 1 ) whose sum is k is given by, while the number of ways to write where every a < sub > i </ sub > is a nonnegative integer is given by.

# for each line, the number of products ( i. e. the sum of the coefficients ) is equal to.

The reflected light being the sum of the incoming light ( L < sub > i </ sub >) from all directions, multiplied by the surface reflection and incoming angle.

The sum of the two values, i. e. the total number of pips, may be referred to as the rank or weight of a tile, and a tile with more pips may be called heavier than a lighter tile with fewer pips.

One lemma states that the difference of the cubes of two rational numbers is equal to the sum of the cubes of two other rational numbers, i. e. given any a and b, with a > b, there exist c and d, all positive and rational, such that

Microeconomics theory considers the aggregates ( the sum of ) of quantity demanded by buyers and quantity supplied by sellers, studying each possible price per unit ( i. e. supply and demand ).

For N interacting particles, i. e. particles which interact mutually and constitute a many-body situation, the potential energy function V is not simply a sum of the separate potentials ( and certainly not a product, as this is dimensionally incorrect ).

For non-interacting particles, i. e. particles which do not interact mutually and move independently, the potential of the system is the sum of the separate potential energy for each particle, that is

If R is the direct sum of the rings R < sub > 1 </ sub >,..., R < sub > n </ sub >, then the identity elements of the rings R < sub > i </ sub > are central idempotents in R, pairwise orthogonal, and their sum is 1.

If a decomposition exists with each c < sub > i </ sub > a centrally primitive idempotent, then R is a direct sum of the corner rings c < sub > i </ sub > Rc < sub > i </ sub >, each of which is ring irreducible.

In particular, solutions can be expressed as the sum of a uniform flow ( i. e. a steady flow equal to the free stream velocity ) plus a free vortex flow, i. e. a circular flow around the airfoil with the speed inversely proportional to the radius.

Every ordered field is a formally real field, i. e., 0 cannot be written as a sum of nonzero squares.

Equivalently, a perfect number is a number that is half the sum of all of its positive divisors ( including itself ) i. e. σ < sub > 1 </ sub >( n ) = 2n.

generated R-module, then is a direct sum of cyclic modules, i. e., modules with one generator.