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.

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 sum of zero items, i. e., the summation of the empty set )

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 ".

The measured force, H, in cutting removal of coatings from their substrates consequently can be seen to be the sum of that force required to cut the coating, Af, that due to the bluntness of the knife, Af, and that due to the friction between the bottom of the knife and the substrate, Af, or Af.

For the first of these, he raised a large sum and a substantial force, arriving in Damietta on 24 October 1249, after the town had already been captured.

The total contact force on the particular internal surface is then expressed as the sum ( surface integral ) of the contact forces on all differential surfaces:

Thus the analysis using Newton's laws of motion can proceed as if the reference frame was inertial, provided the fictitious force terms are included in the sum of external forces.

This force is the sum of an outward centrifugal force and an inward Coriolis force.

The force F is the vector sum of all " real " forces on the particle, such as electromagnetic, gravitational, nuclear and so forth.

The acceleration of each body is equal to the sum of the gravitational forces on it, divided by its mass, and the gravitational force between each pair of bodies is proportional to the product of their masses and decreases inversely with the square of the distance between them.

The sum of force and moment is zero.

Varignon's theorem states that the moment of a force about any point is equal to the sum of the moments of the components of the force about the same point.

The total force on particle k is the sum of all the forces from the other particles j in the system

The number of military personnel in Yemen is relatively high ; in sum, Yemen has the second largest military force on the Arabian Peninsula after Saudi Arabia.

Equivalently, if a particle travels in a closed loop, the net work done ( the sum of the force acting along the path multiplied by the distance travelled ) by a conservative force is zero.

In physical chemistry, the van der Waals force ( or van der Waals interaction ), named after Dutch scientist Johannes Diderik van der Waals, is the sum of the attractive or repulsive forces between molecules ( or between parts of the same molecule ) other than those due to covalent bonds, the hydrogen bonds, or the electrostatic interaction of ions with one another or with neutral molecules.

For macroscopic bodies with known volumes and numbers of atoms or molecules per unit volume, the total van der Waals force is often computed based on the " microscopic theory " as the sum over all interacting pairs.

* An introductory description of the van der Waals force ( as a sum of attractive components only )

The direction of the lever arm holding the mass will be along the vector sum of the reactive centrifugal force vector and the gravitational force.

The center of pressure is the point on a body where the total sum of a pressure field acts, causing a force and no moment about that point.

Drain off the water at such a time that the wheel has become substantially empty of water at the end of the sum of the times shown in Columns A and C, measured from the time the wash wheel was started.

Drain off the water at such a time that the wheel has become substantially empty of water at the end of the sum of the times shown in Columns A, C, and E, measured from the time the wash wheel was started.

Martians use the merging of their bodies with water as a simple example or symbol of how two entities can combine to create a new reality greater than the sum of its parts.

All of the separate chemical compounds, minerals and water — which when piled together constitute the sum total of the actual physical matter of the human body — are not of themselves a human body, however much they may be physically compounded and mixed and rearranged in the laboratory, since they are still only a pile of organic chemicals, minerals and water in a particular complex configuration.

Subtracting the last reading from the sum of the first two readings gives the weight of the displaced water.

The resulting thrust pushes the boat in the opposite direction to the sum of the momentum change in the water flowing through the propeller.

The great king Yu () tried to channel the water out to sea where then emerged from the water a turtle with a curious figure / pattern on its shell ; circular dots of numbers which were arranged in a three by three grid pattern such that the sum of the numbers in each row, column and diagonal was the same: 15, which is also the number of days in each of the 24 cycles of the Chinese solar year.

It refers to the sum of the water use in the various steps of the production chain.

This is the sum of internal renewable water resources and incoming flow originating outside the country.

: And sum were in water adreynt,

* Pressure-the overall pressure on a diver is the sum of the atmospheric pressure and the water pressure.

The German Federal Environmental Agency issued guidelines for the sum of PFOA and PFOS concentrations in drinking water: 0. 1 parts per billion for precaution and 0. 3 parts per billion for a threshold.

* Specific enthalpy, symbolized by h, is the sum of the internal ( heat ) energy of the moist air in question, including the heat of the air and water vapor within.

Many different factors may affect the total water potential, and the sum of these potentials determines the overall water potential and the direction of water flow: