In this case the Selberg trace formula is formally similar to the explicit formulas relating the zeros of the Riemann zeta function to prime numbers, with the zeta zeros corresponding to eigenvalues of the Laplacian, and the primes corresponding to geodesics.

The archives of the pontifices were kept here, the formulas of all kinds of prayers, vows, sacrifices, etc., the state calendar of sacred days, the Annales-the record of events of each year for public reference-and the laws relating to marriage, death, wills, etc.

However, invalid formulas ( those that are not entailed by a given theory ), cannot always be recognized.

SAT is easier if the formulas are restricted to those in disjunctive normal form, that is, they are disjunction ( OR ) of terms, where each term is a conjunction ( AND ) of literals ( possibly negated variables ).

They consist of chemical or structural formulas of the reactants on the left and those of the products on the right.

In those oblique coordinate systems the computations of distances and angles is more complicated than in standard Cartesian systems, and many standard formulas ( such as the Pythagorean formula for the distance ) do not hold.

Other Möbius inversion formulas are obtained when different locally finite partially ordered sets replace the classic case of the natural numbers ordered by divisibility ; for an account of those, see incidence algebra.

In their wake, pop music filled the radio waves, dominated by teen idol crooners who sang cleansed formulas like those about the twist dance craze and " death discs " like " Teen Angel " and " Endless Sleep ".

The main concepts are those of a grid of cells, called sheet, with either raw data, called values, or formulas in the cells.

Also, the charters and documents produced for the government in Normandy differed in formulas from those produced in England.

The path loss in other frequency bands ( MW, SW, Microwave ) is predicted with similar methods, though the concrete algorithms and formulas may be very different from those for VHF / UHF.

; they point out that the traditional formulas with a logarithmic region and a linear region do not fit the data from Stevens and Volkman's curves as well as some other forms, based on the following data table of measurements that they made from those curves:

Among those at the film's premiere was film critic Bosley Crowther of The New York Times, who noted that " motion history was made last night ... Fantasia dumps conventional formulas overboard and reveals the scope of films for imaginative excursion ... Fantasia ... is simply terrific.

* such that the new coordinates of the image of each point are the same as the old coordinates of the original point ( the formulas for the mapping are the inverse of those for the coordinate transformation )

* such that the old coordinates of the image of each point are the same as the new coordinates of the original point ( the formulas for the mapping are the same as those for the coordinate transformation )

While art was creative, Broch held that kitsch depended solely on plundering creative art by adopting formulas that seek to imitate it, limiting itself to conventions and demanding a totalitarianism of those recognizable conventions.

By applying those vectors he significantly simplified the calculation and directly obtained all the formulas of the classical theory of secular perturbations.

This clearly excludes the possibility of having any formula that expresses the solutions of an arbitrary equation of degree 5 or higher in terms of its coefficients, using only those operations, or even of having different formulas for different roots or for different classes of polynomials, in such a way as to cover all cases.

( In principle one could imagine formulas using irrational numbers as constants, but even if a finite number of those were admitted at the start, not all roots of higher-degree equations could be obtained.

There are many different formulas, including waterproof for those prone to allergies or sudden tears.

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.

The present Roman Missal has common formulas for the Dedication of Churches, the Blessed Virgin Mary, Martyrs ( with special formulas for missionary martyrs and virgin martyrs ), Pastors ( subdivided into bishops, generic pastors, founders of churches, and missionaries ), Doctors of the Church, Virgins, and ( generic ) Saints ( with special formulas for abbots, monks, nuns, religious, those noted for works of mercy, educators, and women saints ).

The other difference formulas, such as those of Stirling, Bessel and Gauss, can be derived from Newton's, using Newton's terms, with data points and x values renamed in keeping with the choice of x zero, and based on the fact that they must add up to the same sum value as Newton's ( With Stirling that is so when polynomial degree is even.

Couper adopted the atomic weight of oxygen as 8 rather than 16, so there are twice as many oxygen atoms in Couper's formulas as in those of Kekulé.

The truth values of logical formulas usually form a finite set, generally restricted to two values: true and false, but logic can also be continuous-valued, e. g., fuzzy logic.

The two equations above can be derived by adding or subtracting Euler's formulas:

There are two key types of legal expressions: terms, which intuitively represent objects, and formulas, which intuitively express predicates that can be true or false.

And since these two formulas are equivalent, if we replace the first with the second inside Φ, we obtain the formula Φ ' such that Φ ≡ Φ ':

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

( In Principia Mathematica, formulas and propositions are identified by a leading asterisk and two numbers, such as "✸ 2. 1 ".

However with the help of the correct atomic weights for copper and oxygen ( which Dalton did not know ), it is determined that the formulas of these two oxides are Cu < sub > 2 </ sub > O and CuO.

The differences between tantalum and niobium were unequivocally demonstrated in 1864 by Christian Wilhelm Blomstrand, and Henri Etienne Sainte-Claire Deville, as well as Louis J. Troost, who determined the formulas of some of the compounds in 1865 and finally by the Swiss chemist Jean Charles Galissard de Marignac in 1866, who all proved that there were only two elements.

The two professors in Christiania, Søren Rasmussen and Christopher Hansteen, found no errors in Abel's formulas, and sent the work on to the leading mathematician in the Nordic countries, Professor Ferdinand Degen in Copenhagen.

This includes 12 years in a row from 1981 – 92 – all as a result of the scheduling formulas in place prior to 2002 ( this remains a record for most consecutive years in which two teams not from the same division met each other ).

The following two formulas involve quickly converging series, and are thus appropriate for numerical computation:

Darby first cites F. W. Maitland's comment following his compilation of a table of statistics from material taken from the Domesday Book survey, " it will be remembered that, as matters now stand, two men not unskilled in Domesday might add up the number of hides in a county and arrive at very different results because they would hold different opinions as to the meanings of certain formulas which are not uncommon ", then after adding that " each county presents its own problems " Darby concedes that " it would be more correct to speak not of ' the Domesday geography of England ', but of ' the geography of Domesday Book '.

The PRC's One-China policy rejects formulas which call for " two Chinas " or " one China, one Taiwan " and has stated that efforts to divide the sovereignty of China could be met with military force.

Quadratic polynomials can sometimes be factored into two binomials with simple integer coefficients by use of Vieta's formulas, without the need to use the quadratic formula.

If we now equate our two formulas for we obtain:

< li > We will compute certain multiples ( k times ) of the point P using the standard addition rule on our elliptic curve: given two points P, Q on the curve, their sum can be computed using the formulas given in the group law section of the article on elliptic curves.

The two formulas can be combined to determine the present value of the bond.

The claim that two formulas are logically equivalent is a statement in the metalanguage, expressing a relationship between two statements p and q.

Hence, high-level financial inventory has these two basic formulas, which relate to the accounting period:

Heron's formula is distinguished from other formulas for the area of a triangle, such as half the base times the height or half the modulus of a cross product of two sides, by requiring no arbitrary choice of side as base or vertex as origin.

Gödel specifically used this scheme at two levels: first, to encode sequences of symbols representing formulas, and second, to encode sequences of formulas representing proofs.

Soon Luria had two classes of disciples: ( 1 ) novices, to whom he expounded the elementary Kabbalah, and ( 2 ) initiates, who became the depositories of his secret teachings and his formulas of invocation and conjuration.