A remarkable early tenth-century document, known as the Burghal Hidage, provides a formula for determining how many men were needed to garrison a borough, based on one man for every of wall.

Euler's formula provides a means of conversion between cartesian coordinates and polar coordinates.

Euler's formula provides a powerful connection between analysis and trigonometry, and provides an interpretation of the sine and cosine functions as weighted sums of the exponential function:

In mathematics, the Euler – Maclaurin formula provides a powerful connection between integrals ( see calculus ) and sums.

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

Kirchhoff's diffraction formula provides a rigorous mathematical foundation for diffraction, based on the wave equation.

This provides a simple formula for the mechanical advantage of the system.

When the curve is given as a continuous function, perhaps as an algebraic formula, then the differential calculus provides rules giving a formula for the slope of the curve at any point in the middle of the curve.

When the formula is applied to each lifter's overall total and then grouped along with the other competitors ' and evaluated, it provides a numeric result which determines the competition's best overall men's and women's lifters.

An alternative formula is a single chief political body ( e. g. presidium ) which collectively leads the government and provides ( e. g. by turns ) the ceremonial Head of state

This formula provides an approximation with fair accuracy for most people, assuming the blood was drawn after fasting for about 14 hours or longer.

In the case when the coefficient ring is a field of characteristic 0, the Lagrange inversion formula provides a powerful tool to compute the coefficients of g, as well as the coefficients of the ( multiplicative ) powers of g.

However, this formula still provides a good approximation for long thin conductors such as wires.

This equation provides a formula for the velocity of B in terms of the velocity of A and its relative velocity,

In the latter case, it provides an alternative formula for expressing local relationships between distances:

where the temperature offset of 6 kelvins provides the formula with a better fit to reality at lower temperatures.

Another relationship is given by the Gödel – Gentzen negative translation, which provides an embedding of classical first-order logic into intuitionistic logic: a first-order formula is provable in classical logic if and only if its Gödel – Gentzen translation is provable intuitionistically.

Thus in most " natural " settings, if a complex number provides a solution to a problem, so does its conjugate, such as is the case for complex solutions of the quadratic formula with real coefficients.

In the following example, the formula R ( 3, 3 ) provides a solution to the question which asks the minimum number of vertices a graph must contain in order to ensure that either ( 1 ) at least 3 vertices in the graph are connected or ( 2 ) at least 3 vertices in the graph are unconnected.

The Tarski-Kuratowski algorithm provides an easy way to get an upper bound on the classifications assigned to a formula and the set it defines.

" This formula provides little instruction to an ordinary decision maker.

Given a simple polygon constructed on a grid of equal-distanced points ( i. e., points with integer coordinates ) such that all the polygon's vertices are grid points, Pick's theorem provides a simple formula for calculating the area A of this polygon in terms of the number i of lattice points in the interior located in the polygon and the number b of lattice points on the boundary placed on the polygon's perimeter:

For example, in a system where there is no queuing, the GoS may be that no more than 1 call in 100 is blocked ( i. e., rejected ) due to all circuits being in use ( a GoS of 0. 01 ), which becomes the target probability of call blocking, P < sub > b </ sub >, when using the Erlang B formula.

For example, if a person stole the Coca-Cola formula, the crime would be larceny but the grade of the offense would be determined by the value of the paper on which the formula was recorded not the value of the recipe.

The grade level is calculated with the following formula:

To compensate for this, the gradient should be a little less steep the sharper the curve is ; the necessary grade reduction is assumed to be given by a simple formula such as 0. 04 per cent per " degree of curve ", the latter being a measure of curve sharpness used in the United States.

This formula is used to determine the number of agents or customer service representatives needed to staff a call centre, for a specified desired probability of queuing.

His Pantheisticon, sive formula celebrandae sodalitatis socraticae ( Pantheisticon, or the Form of Celebrating the Socratic Society ), of which he printed a few copies for private circulation only, gave great offence as a sort of liturgic service made up of passages from pagan authors, in imitation of the Church of England liturgy.

He created the editorial formula still in use today, brought aboard Steven Slon, a veteran of the service journalism category, imported Greg Gutfeld from Rodale's Prevention title and worked with longtime staff editor Denis Boyles, a former Playboy contributing editor, to develop the magazine's influential and distinctive " regular guy " voice.

They promote, develop, and expand school-based service learning to K – 12 schools and school districts ; they provide direct assistance in the form of technical support and professional development opportunities to local school-community partnerships ; and they administer and disseminate the annual K – 12 school-based state formula grants from Learn and Serve America, the primary federal funding source for service learning.

Virginia soldiers of the Continental line, who served in the Revolutionary War, were eligible to procure a bounty award in the form of land, according to a formula based on rank and time of service.

Some Lutherans use this formula as their rationale for opposing in the church the reservation of the consecrated elements, private masses, the practice of Corpus Christi, and the belief that the reliquæ ( what remains of the consecrated elements after all have communed in the worship service ) are still sacramentally united to the Body and Blood of Christ.

The monocoque BRM P261 V8 car was soon developed and these ran on through the 1. 5-litre formula and performed useful service in the early races of the subsequent 3. 0-litre formula.

This service could not be equaled by competing steam railroads, and for a few years it was a successful formula.

A defined benefit ( or pension ) plan calculates benefits using a fixed formula that typically factors in final pay and service with an employer, and payments are made from a trust fund specifically dedicated to the plan.

Funding is based on a complex formula that includes population, calls for service, and felony crimes in the prior year.

Overall benefits paid to CSRS or CSRS Offset employees will remain equitable based on the number of years of creditable service and CSRS formula upon retirement.

With the formula C < sub > n </ sub > H < sub > 2n + 2 </ sub >, Alkanes belong to a homologous series of organic compounds in which the members differ by a constant relative molecular mass of 14.

An important class of alcohols are the simple acyclic alcohols, the general formula for which is C < sub > n </ sub > H < sub > 2n + 1 </ sub > OH.

which means that we have analytically derived a formula for the output alpha and the output color of.

The final speedup is computed by dividing the old running time by the new running time, which is what the above formula does.

Gilmore's program used conversion to disjunctive normal form, a form in which the satisfiability of a formula is obvious.

His early work in scrutinizing the use of local, exchange and trunk telephone line usage in a small community, to understand the theoretical requirements of an efficient network led to the creation of the Erlang formula, which became a foundational element of present day telecommunication network studies.

A general formula by which some of these numbers could be derived was invented circa 850 by Thābit ibn Qurra ( 826-901 ).

The simplest type of carbohydrate is a monosaccharide, which among other properties contains carbon, hydrogen, and oxygen, mostly in a ratio of 1: 2: 1 ( generalized formula C < sub > n </ sub > H < sub > 2n </ sub > O < sub > n </ sub >, where n is at least 3 ).

In calculations of the maximum symbol rate, the Nyquist sampling rate, and maximum bit rate according to the Hartley formula, the bandwidth refers to the frequency range within which the gain is non-zero, or the gain in dB is below a very large value.

Chapters 1: 1-2: 5 are thus a confession of failure ; chapters 2: 6-3: 6 are a major summary and reflection from the Deuteronomists, setting out the over all formula which the stories in the main text will follow: Israel " does evil in the eyes of Yahweh ;" the people are given into the hands of their enemies and cry out to Yahweh ; Yahweh raises up a leader ; the " spirit of Yahweh " comes upon the leader, the enemy is defeated, and peace is regained.

An identical expression to Einstein's formula for the diffusion coefficient was also found by Walther Nernst in 1888 in which he expressed the diffusion coefficient as the ratio of the osmotic pressure to the ratio of the frictional force and the velocity to which it gives rise.

He writes for the diffusion coefficient, where is the osmotic pressure and is the ratio of the frictional force to the molecular viscosity which he assumes is given by Stokes's formula for the viscosity.

At first the predictions of Einstein's formula were seemingly refuted by a series of experiments by Svedberg in 1906 and 1907, which gave displacements of the particles as 4 to 6 times the predicted value, and by Henri in 1908 who found displacements 3 times greater than Einstein's formula predicted.

If one integrates this picture, which corresponds to applying the fundamental theorem of calculus, one obtains Cavalieri's quadrature formula, the integral – see proof of Cavalieri's quadrature formula for details.

Equally important is to determine whether no such assignments exist, which would imply that the function expressed by the formula is identically FALSE for all possible variable assignments.

For example, the formula a AND b is satisfiable because one can find the values a = TRUE and b = TRUE, which make ( a AND b ) = TRUE.

The propositional satisfiability problem ( PSAT ), which decides whether a given propositional formula is satisfiable, is of central importance in various areas of computer science, including theoretical computer science, algorithmics, artificial intelligence, hardware design, electronic design automation, and verification.

There are several special cases of the Boolean satisfiability problem in which the formula are required to be conjunctions of clauses ( i. e. formulae in conjunctive normal form ).

Determining the satisfiability of a formula in which each clause is limited to at most two literals is NL-complete ; this problem is called " 2SAT ".

Determining the satisfiability of a formula in which each clause is a Horn clause ( i. e. it contains at most one positive literal ) is P-complete ; this problem is called Horn-satisfiability.

If we allow both, the problem is called the quantified Boolean formula problem ( QBF ), which can be shown to be PSPACE-complete.

MAJSAT, which asks if the majority of all assignments make the formula true, is complete for PP, a probabilistic class.