To sum things up, the use of money in trading all goods ( capital / labor and consumer ) in all markets ( spot and financial ) combined with profit driven entrepreneurship and Darwinian natural selection in financial markets all combine to make rational economic calculation and allocation the outcome of the capitalist process.

This probably convinced him that the sum equals π < sup > 2 </ sup > / 6, which he proved in the same year.

Earlier Berkeley studies estimated that by the end of 1999, the sum of human-produced information ( including all audio, video recordings and text / books ) was about 12 exabytes of data.

where D / Dt is the substantial derivative, which is the sum of local and convective derivatives.

* nautā ( ablative ) " by / with / from / in sailor " various uses not covered by the above ( e. g. sum altior nautā I am taller than the sailor ).

For different situations and / or number of particles, the Hamiltonian is different since it includes the sum of kinetic energies of the particles, and the potential energy function corresponding to the situation.

A substantial simplification proposes that other forms of energy changes are inconsequential during the flow of air around a wing and that energy transfer in / out of the air is not significant, so the sum of pressure energy and speed energy for any particular parcel of air must be constant.

Wintermute / Neuromancer contacts him, saying that it has become " the sum total of the works, the whole show ," and has begun looking for other AIs like itself.

: Fyrigev okkum syndir okkara, / so sum vit eisini fyrigeva teimum, ið móti okkum synda.

To get the percentage, we add $ 30 and $ 10 to get a sum of $ 40 and then divide $ 10 by $ 40, giving us 1 / 4, or 25 %.

Equivalently, the number 6 is equal to half the sum of all its positive divisors: ( 1 + 2 + 3 + 6 ) / 2 = 6.

All legionary soldiers would also receive a sizeable sum of money on the completion of their term of service: 3000 denarii from the time of Augustus and / or a plot of good farmland ( good land was in much demand ); farmland given to veterans often helped in establishing control of the frontier regions and over rebellious provinces.

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.

Rotary wing aircraft and thrust vectoring V / STOL aircraft use engine thrust to support the weight of the aircraft, and vector sum of this thrust fore and aft to control forward speed.

In the limit, as n goes to infinity, the limit of the sum of the powers of 4 / 9 is 4 / 5, so

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.

* A ring is ( left / right ) Noetherian if and only if every direct sum of injective ( left / right ) modules is injective.

It is ideal for a bookmaker to price / mark up a book such that the net outcome will always be in his favour, i. e. the sum of the probabilities quoted for all possible outcomes will be in excess of 100 %.

As a consequence, if f and g are as above, with, we can relate the complex powers of f / X and g / X: precisely, if and are non-zero complex numbers with negative integer sum,, then

The algorithm can be extended to handle unequal sample weights, replacing the simple counter n with the sum of weights seen so far.

and mean that the sum or product is over all positive divisors of n, including 1 and n.

The notations can be combined: and mean that the sum or product is over all prime divisors of n.

and similarly and mean that the sum or product is over all prime powers dividing n.

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.

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.

Bernoulli numbers feature prominently in the closed form expression of the sum of the m-th powers of the first n positive integers.

Equivalently, the determinant can be expressed as a sum of products of entries of the matrix where each product has n terms and the coefficient of each product is − 1 or 1 or 0 according to a given rule: it is a polynomial expression of the matrix entries.

The Euler – Maclaurin formula provides expressions for the difference between the sum and the integral in terms of the higher derivatives ƒ < sup >( k )</ sup > at the end points of the interval m and n. Explicitly, for any natural number p, we have

It states that for all integers n and k greater than 1, if the sum of n kth powers of positive integers is itself a kth power, then n is greater than or equal to k.

The integer sequence of the number of idempotent functions as given by the sum above for n = 0, 1, 2, ... starts with 1, 1, 3, 10, 41, 196, 1057, 6322, 41393, ...

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.

Conversely, given central idempotents a < sub > 1 </ sub >,..., a < sub > n </ sub > in R that are pairwise orthogonal and have sum 1, then R is the direct sum of the rings Ra < sub > 1 </ sub >,…, Ra < sub > n </ sub >.

Let ε ( n ) denote the energy of a particle in state n. As the particles do not interact, the total energy of the system is the sum of the single-particle energies.

* http :// www. gendisasters. com / data1 / mn / fires / minneapolis-mill-fire1881. htm

: Precise instructions ( in language understood by " the computer ") for a fast, efficient, " good " process that specifies the " moves " of " the computer " ( machine or human, equipped with the necessary internally contained information and capabilities ) to find, decode, and then process arbitrary input integers / symbols m and n, symbols + and =

sum1 / n

( sum2-sum3 ** 2 / n )/( n-1 )

mean + delta / n

M2 / n # Population Variance

( n * M4 ) / ( M2 * M2 )-3

covariance += a * b / n

n / a

* Any ring of characteristic n is a ( Z / nZ )- algebra in the same way.

S ( a, b )=( nc + me / m + n, nd + mf / m + n )

The year 1 BC / BCE is numbered 0, the year 2 BC is numbered − 1, and in general the year n BC / BCE is numbered "−( n − 1 )" ( a negative number equal to 1 − n ).

The numbers of AD / CE years are not changed and are written with either no sign or a positive sign ; thus in general n AD / CE is simply n or + n.