These formulas can even serve as the definition of the trigonometric functions for complex arguments x.

These results, however, are very sensitive to the accuracy of the twiddle factors used in the FFT ( i. e. the trigonometric function values ), and it is not unusual for incautious FFT implementations to have much worse accuracy, e. g. if they use inaccurate trigonometric recurrence formulas.

These simple formulas will not do if the input values are not equally likely, or are not independent.

These two researchers developed several formulas for mineral nutrient solutions, known as Hoagland solution.

These same interpolation formulas nevertheless continue to be used as part of the software algorithms for solving differential equations.

These formulas provide the solution for the initial-value problem for the wave equation.

These changes can be represented in formulas by inserting appropriate electrons into each half-reaction:

These formulas however only provide the calculation for the τ = τ < sub > 0 </ sub > case.

These formulas may be used when h is much smaller than the radius of the Earth ( 6371 km ), including all views from any mountaintops, aeroplanes, or high-altitude balloons.

These formulas characterize the periapsis and apoapsis of an orbit:

These formulas are traditionally used in analog televisions and equipment ; digital equipment such as HDTV and digital video cameras use Y ' CbCr.

These formulas are connected to the Schubert decomposition of the Grassmannian, and are q-analogs of the Betti numbers of complex Grassmannians.

These formulas are the basis for comparing and investigating the nature of non-random patterns in the structure of food web networks among many different types of ecosystems.

These formulas are generalized by the Faà di Bruno's formula.

These formulas can also be written using Lagrange's notation.

These formulas contain the " slope " of the line connecting P and Q, hence involve divisions between residue classes modulo n, which can be performed using the extended Euclidean algorithm.

These operators are direct extensions of their complex analogs: if and are taken from the real subset of complex numbers, the appearance of the conjugate in the formulas has no effect, so the operators are the same as those for the complex numbers.

These ideas were regarded as dynamic forces that Herbart attempted to explain by means of mathematical formulas.

These formulas for denatured alcohol are found in 27 CFR part 21 of the Code of Federal Regulations.

These formulas are for single-frequency sine waves or narrowband signals.

These formulas only state in another form the assertions in the table of derivatives.

These formulas all provide a different view into the processes that generate longshore drift.

These five formulas are:

These formulas give the explicit numbers of rational curves of all degrees in a large class of Calabi – Yau manifolds, in terms of the Picard – Fuchs equations of the corresponding mirror manifolds.

These two formulas, added together, produce

These differences in turn result from the fact that my Yokuts vocabularies were built up of terms selected mainly to insure unambiguity of English meaning between illiterate informants and myself, within a compact and uniform territorial area, but that Hoijer's vocabulary is based on Swadesh's second glottochronological list which aims at eliminating all items which might be culturally or geographically determined.

These are arranged approximately in the order of the vociferousness of their proponents but will be discussed in the reverse order in the hope that the true order of importance will result.

" These result from a direct contact with nature, and through nature a closer relationship to God.

These displayed his prolific habits and literally encyclopedic knowledge of topics such as logic, theology, botany, geography, astronomy, astrology, mineralogy, chemistry, zoology, physiology, phrenology and others ; all of which were the result of logic and observation.

These critics assert that people from the Third World see the anti-globalization movement as a threat to their jobs, wages, consuming options and livelihoods, and that a cessation or reversal of globalization would result in many people in poor countries being left in greater poverty.

These changes maintain the person with alcoholism's compulsive inability to stop drinking and result in alcohol withdrawal syndrome if the person stops.

These high-priority agents pose a risk to national security, can be easily transmitted and disseminated, result in high mortality, have potential major public health impact, may cause public panic, or require special action for public health preparedness.

These procedures are preferred by some breast cancer patients because they result in an abdominoplasty ( tummy tuck ), and allow the breast to be reconstructed with one's own tissues instead of a foreign implant.

These changes came about partly as the result of the urgings of Edwin Samuel Montagu, an influential anti-Zionist Jew and secretary of state for India, who was concerned that the declaration without those changes could result in increased anti-Semitic persecution.

These may be brought about, for example, by such factors as poor management, lack of consultation with employees, personality conflicts which can result in people delaying or refusing to communicate, the personal attitudes of individual employees which may be due to lack of motivation or dissatisfaction at work, brought about by insufficient training to enable them to carry out particular tasks, or just resistance to change due to entrenched attitudes and ideas.

These may result from individuals ' personal discomfort, caused — for example — by ill health, poor eyesight or hearing difficulties.

These are often accepted as random selections because the number of words has not been calculated beforehand, so the result is unknown right up until someone is selected.

These were partly the result of the tireless travels he had begun in 1839 in search of unread manuscripts of the New Testament, " to clear up in this way ," he wrote, " the history of the sacred text, and to recover if possible the genuine apostolic text which is the foundation of our faith.

These perturbations result in different arrangement of electrons in the excited state of the involved molecules and therefore in different effects.

These closures will result in total job losses of 600 in Georgia.

These uniting congregations were themselves the result of several previous unions.

These myths often depict creation as the result of a sexual union, and serve as genealogical record of the deities born from it.

These result from a rapid myelinolysis of the corticobulbar and corticospinal tracts in the brainstem.

These imports declined sharply to an average of only US $ 50 million annually between 1985 and 1987, presumably as a result of a dramatic reduction in oil revenues and the precipitous drop in the value of the sucre, which made imported arms extremely expensive.

These processes may result in artifacts, but these can usually be identified by comparing the results obtained by using radically different specimen preparation methods.

These measurements are then used to calculate a single result, most commonly the busy hour traffic ( in erlangs ).

These changes taken together have been interpreted as a result of an increased emphasis on pair bonding as a possible solution to the requirement for increased parental investment due to the prolonged infancy of offspring.

These qualities were alleged against him as defects, and to prove that his style was not the result of want of power, he painted the magnificent figure of the St. Mark Evangelist ( ranked as his masterpiece ), and the undraped figure of Saint Sebastian.

These reciprocal effects result in gene environment correlation.