The number of degrees of freedom DF can be partitioned in a similar way: one of these components ( that for error ) specifies a chi-squared distribution which describes the associated sum of squares, while the same is true for " treatments " if there is no treatment effect.

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.

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

The wave displacement at any subsequent point is the sum of these secondary waves.

They might have some value, but when we consider the total value of a consciousness experiencing a beautiful object, it seems to exceed the simple sum of these values ( Principia 18: 2 ).

In 1678, Huygens proposed that every point to which a luminous disturbance reaches becomes a source of a spherical wave ; the sum of these secondary waves determines the form of the wave at any subsequent time.

In the equation for Z, every possible permutation of the ns occurs once in the sum, even though each of these permutations is describing the same multi-particle state.

Again, on average the worker can transfer no more than the value of these means of labor previously possessed to the finished cup of coffee So the value of coffee produced in a day equals the sum of both the value of the means of labor — this constant capital — and the value newly added by the worker in proportion to the duration and intensity of their work.

Louis purchased these in 1239 – 41 from Emperor Baldwin II of the Latin Empire of Constantinople, for the exorbitant sum of 135, 000 livres ( the chapel, on the other hand, cost only 60, 000 livres to build ).

The thermal plasma within the inner magnetosphere corotates with the Earth and therefore reacts to the sum of these two fields.

In word-final position these all contrast, as shown by the minimal triplet sum, sun, sung.

To convert this ratio to the equivalent percentage, we add these two numbers together and then divide the cost of the call by this sum.

She is the primordial energy that holds all creation and destruction, all cycles of birth and death, all laws of cause and effect within Herself, and yet is greater than the sum total of all these.

As one wraps around the polygon, these triangles with positive and negative area will overlap, and the areas between the origin and the polygon will be cancelled out and sum to 0, while only the area inside the reference triangle remains.

There is a restriction that these probabilities sum to 1.

Once this was understood, the various reign lengths and cross-synchronisms for these kings was worked out, and the sum of reigns for both kingdoms produced 931 / 930 BC for the division of the kingdom when working backwards from the Battle of Qarqar in 853 BC.

) The sum of these characteristics gives a total of Hero Points used to buy super powers.

In sum these measures were seen as the re-transformation of proletarians within production from collective subject back into the atomised objects of capital.

The sum of these time differences is the round trip delay between the two nodes.

Since commercials are not sold in these markets, Producers Sales Organization lost an undisclosed sum of money.

Taking the sum of these two forces and dividing with the mass one gets for the vertical motion:

The emergent is unlike its components insofar as these are incommensurable, and it cannot be reduced to their sum or their difference.

The scaled and time-shifted sinc functions are continuous making the sum of these also continuous, so the result of this operation is a continuous signal.

In a convex combination, each point in is assigned a weight or coefficient in such a way that the coefficients are all positive and sum to one, and these weights are used to compute a weighted average of the points.

One can then evaluate the probability amplitude at the photon's final point ; next, one can integrate over all of these arrows ( see vector sum ), and square the length of the result to obtain the probability that this photon will reflect off of the mirror in the pertinent fashion.

Similarly, if light consisted strictly of classical particles and we illuminated two parallel slits, the expected pattern on the screen would simply be the sum of the two single-slit patterns.

The parallel postulate: If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

It is similar to a first-order chemical reaction in which the first-order rate constant is the sum of all of the rates ( a parallel kinetic model ).

Normal play Nim ( or more precisely the system of nimbers ) is fundamental to the Sprague-Grundy theorem, which essentially says that in normal play every impartial game is equivalent to a Nim heap that yields the same outcome when played in parallel with other normal play impartial games ( see disjunctive sum ).

In this geometry, an infinite number of parallel lines pass through the point P. Consequently the sum of angles in a triangle is less than 180 ° and the ratio of a circle's circumference to its diameter is greater than pi.

After many failed attempts to derive the parallel postulate from other axioms, the study of the still hypothetical hyperbolic geometry by Johann Heinrich Lambert ( 1728 – 1777 ) led him to introduce the hyperbolic functions and compute the area of a hyperbolic triangle ( where the sum of angles is less than 180 °).

Later in the 19th century, the German mathematician Bernhard Riemann developed Elliptic geometry, another non-Euclidean geometry where no parallel can be found and the sum of angles in a triangle is more than 180 °.

A rhombus therefore has all of the properties of a parallelogram: for example, opposite sides are parallel ; adjacent angles are supplementary ; the two diagonals bisect one another ; any line through the midpoint bisects the area ; and the sum of the squares of the sides equals the sum of the squares of the diagonals ( the parallelogram law ).

* Book 1 contains Euclid's 10 axioms ( 5 named postulates — including the parallel postulate — and 5 named axioms ) and the basic propositions of geometry: the pons asinorum ( proposition 5 ), the Pythagorean theorem ( Proposition 47 ), equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are " equal " ( have the same area ).

In a parallel circuit, the voltage across each of the components is the same, and the total current is the sum of the currents through each component.

For N equal resistors in parallel, the reciprocal sum expression simplifies to:

Inductors follow the same law, in that the total inductance of non-coupled inductors in parallel is equal to the reciprocal of the sum of the reciprocals of their individual inductances:

The total capacitance of capacitors in parallel is equal to the sum of their individual capacitances:

The Guide describes the WSOGMM as a way of understanding the sum of parallel universes in creation ( in this context, creation is best described to a casual observer as spacetime ).

That sum will be performed in parallel with the fused add / decode described here.

It is itself a vector field ; it is the sum of the infinitessimal change in the vector field, as one moves along the direction, plus the infinitessimal change of the vector due to its parallel transport along the direction.

Lobachevsky replaced Euclid's parallel postulate with the statement that for any given point there exists more than one line that can be extended through that point and run parallel to another line of which that point is not part ; a famous consequence is that the sum of angles in a triangle must be less than 180 degrees.

He has also written highly-cited research papers on the NP-completeness of approximately solving certain optimization problems, on open shop scheduling, on parallel algorithms for matrix multiplication and their application in graph theory, and on improved exponential time exact algorithms for the subset sum problem, among his many other research results.

If the initial curve is a boundary of a planar set and its parallel curve is without self-intersections, then the latter is the boundary of the Minkowski sum of the planar set and the disk of the given radius.

Circular angles can be characterised geometrically by the property that the if two chords P < sub > 0 </ sub > P < sub > 1 </ sub > and P < sub > 0 </ sub > P < sub > 2 </ sub > subtend angles L < sub > 1 </ sub > and L < sub > 2 </ sub > at the centre of a circle, their sum L < sub > 1 </ sub > + L < sub > 2 </ sub > is the angle subtended by a chord PQ, where PQ is required to be parallel to P < sub > 1 </ sub > P < sub > 2 </ sub >.

In a diagonal magic cube of order m, all 6m of the diagonals in the m planes parallel to the top, front, and sides of the cube must sum correctly.

For programs executed in parallel, the CPU time will be the sum of CPU times devoted to the task by each CPU running it.