Weber found that the just noticeable difference ( jnd ) between two weights was approximately proportional to the mass of the weights.

In psychophysics, a just-noticeable difference, customarily abbreviated with lowercase letters as jnd, is the smallest detectable difference between a starting and secondary level of a particular sensory stimulus.

For many sensory modalities, over a wide range of stimulus magnitudes sufficiently far from the upper and lower limits of perception, the ' jnd ' is a fixed proportion of the reference sensory level, and so the ratio of the jnd / reference is roughly constant ( that is the jnd is a constant proportion / percentage of the reference level ).

He also found that the just-noticeable difference ( jnd ) of the change in the magnitude of a stimulus ( viz., small weights held in the hand ) is proportional to the magnitude of the stimulus ( e. g., 5 %), rather than being an absolute value ( e. g., 5 grams ).

Thus, if 105 g can ( only just ) be distinguished from 100 g, the jnd ( or differential threshold ) is 5 g. If the mass is doubled, the differential threshold also doubles to 10 g, so that 210 g can be distinguished from 200 g. In this example, a weight ( any weight ) seems to have to increase by 5 % for someone to be able to reliably detect the increase, and this minimum required fractional increase ( of 5 / 100 of the original weight ) is referred to as the " Weber fraction " for detecting changes in weight.

where is the original intensity of stimulation, is the addition to it required for the difference to be perceived ( the jnd ), and k is a constant.

The jnd is a statistical, rather than an exact quantity: from trial to trial, the difference that a given person notices will vary somewhat, and it is therefore necessary to conduct many trials in order to determine the threshold.

The jnd usually reported is the difference that a person notices on 50 % of trials.

If a different proportion is used, this should be included in the description — for example one might report the value of the " 75 % jnd ".

Modern approaches to psychophysics, for example signal detection theory, imply that the observed jnd, even in this statistical sense, is not an absolute quantity, but will depend on situational and motivational as well as perceptual factors.

But it is apparent that no acceptable formula has been found to prevent such a possibility.

One has a recursive formula for binomial coefficients

For example, if a graph has 17 valid 3-colorings, the SAT formula produced by the reduction will have 17 satisfying assignments.

The problem of how many variable assignments satisfy a formula, not a decision problem, is in # P. UNIQUE-SAT or USAT or Unambiguous SAT is the problem of determining whether a formula known to have either zero or one satisfying assignments has zero or has one.

Propositional satisfiability has various generalisations, including satisfiability for quantified Boolean formula problem, for first-and second-order logic, constraint satisfaction problems, 0-1 integer programming, and maximum satisfiability problem.

After one pair of Bézout coefficients ( x, y ) has been computed ( using extended Euclid or some other algorithm ), all pairs may be found using the formula

It has a formula of C < sub > 12 </ sub > H < sub > 22 </ sub > O < sub > 11 </ sub >.

( Some exceptions exist ; for example, deoxyribose, a component of DNA, has the empirical formula C < sub > 5 </ sub > H < sub > 10 </ sub > O < sub > 4 </ sub >.

The aldohexose D-glucose, for example, has the formula ( C · H < sub > 2 </ sub > O ) < sub > 6 </ sub >, of which all but two of its six carbons atoms are stereogenic, making D-glucose one of 2 < sup > 4 </ sup > = 16 possible stereoisomers.

In this situation, the chain rule represents the fact that the derivative of is the composite of the derivative of f and the derivative of g. This theorem is an immediate consequence of the higher dimensional chain rule given above, and it has exactly the same formula.

Church subsequently modified his methods to include use of Herbrand – Gödel recursion and then proved ( 1936 ) that the Entscheidungsproblem is unsolvable: There is no generalized " effective calculation " ( method, algorithm ) that can determine whether or not a formula in either the recursive-or λ-calculus is " valid " ( more precisely: no method to show that a well formed formula has a " normal form ").

It has the following formula under cold air standard analysis:

The formula is more complex than the Otto cycle ( petrol / gasoline engine ) relation that has the following formula ;

Using the DPV formula above ( FV =$ 150, 000, i = 0. 05, n = 3 ), that means that the value of $ 150, 000 received in three years actually has a present value of $ 129, 576 ( rounded off ).

For example the ester hexyl octanoate, also known under the trivial name hexyl caprylate, has the formula CH < sub > 3 </ sub >( CH < sub > 2 </ sub >)< sub > 6 </ sub > CO < sub > 2 </ sub >( CH < sub > 2 </ sub >)< sub > 5 </ sub > CH < sub > 3 </ sub >.

Such a formula for estimating energy typically has relative error of order of 10 %, but can be used to get a rough qualitative idea and understanding of a molecule.

This imbalance has received considerable criticism, and is the subject of 2007 legislative proposals designed to alter the formula.

The Divine Liturgy of St. John Chrysostom, used in the Byzantine Churches, still has a formula of dismissal of catechumens ( not usually followed by any action ) at this point.

For example, the chemical compound n-hexane has the structural formula, which shows that it has 6 carbon atoms arranged in a chain, and 14 hydrogen atoms.

In 1917 Theremin wrote that Ioffe talked of electrons, the photoelectric effect and magnetic fields as parts of an objective reality that surrounds us everyday, unlike others that talked more of somewhat abstract formula and symbols.

The working group had the primary objective of cutting costs to just A $ 250, 000 per car, with a projected timeline of introducing the new formula for the 2013 season.

Washington Allston coined the term " objective correlative ," which T. S. Eliot described as a situation or a chain of events that acts as a formula and is used in art to evoke emotion.

Nevertheless, both a lattice model and the Black – Scholes – Merton formula, as well as other valuation techniques that meet the requirements in paragraph A8, can provide a fair value estimate that is consistent with the measurement objective and fair-value-based method of this Statement.

For a telescope, the angular magnification MA produced by the combination of a particular eyepiece and objective can be calculated with the following formula:

For refutation tableaux, the objective is to show that the negation of a formula cannot be satisfied.

Prior to the fourth plan, the allocation of state resources was based on schematic patterns rather than a transparent and objective mechanism, which led to the adoption of the Gadgil formula in 1969.

The ECJ ruled that ( a ) a chemical formula depicting this scent did not represent the odour of a substance, was not sufficiently intelligible, nor sufficiently clear and precise ; ( b ) a written description was not sufficiently clear, precise and objective ; and ( c ) a physical deposit of a sample of the scent did not constitute a graphic representation, and was not sufficiently stable or durable.

Porter and Duff gave a geometric interpretation of the alpha compositing formula by studying orthogonal coverages.

Neither of these men saw the geometrical interpretation of the formula: the view of complex numbers as points in the complex plane arose only some 50 years later ( see Caspar Wessel ).

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:

The meaning of Logos also is subject to interpretation: " word ", " account ", " plan ", " formula ", " measure ", " proportion ", " reckoning.

A geometric interpretation of Euler's formula

A validity is a formula that is true under any possible interpretation, e. g. in classical propositional logic validities are tautologies.

An interpretation satisfies this formula only if at least one of the new variables is true.

There is some debate as to the informal interpretation of the Barcan formula and its converse.

Furthermore, a formula need not be given any interpretation.

* A formula A in a language is valid if it is true for every interpretation of.

* A formula A in a language is satisfiable if it is true for some interpretation of.

After long and vehement debates, a formula was presented by Bucer and accepted by the majority, so worded as to be capable of bearing a Catholic and a Lutheran interpretation.

The interpretation of this formula is shown in the figure to the right, showing two possible scenarios with acceptable and poor GDOP.

Although there are still obscure points in the interpretation of line two, it is generally accepted that the text contains the formula of an oath.

The interpretation states exactly what is intended to be a proof of a given formula.

Compare this with the standard semantics which says that a universal ( existential ) formula is true if and only if for all ( some ) members of the domain, the formula holds for all ( some ) of them ; e. g. ∀ xA is true ( under an interpretation ) if and only if for all k in the domain D, A ( k / x ) is true ( where A ( k / x ) is the result of substituting k for all occurrences of x in A ).