In other words, if an ' uninhabited ' or ' infidel ' territory is colonized by Britain, then the English law automatically applies in this territory from the moment of colonization ; however if the colonized territory has a pre-existing legal system, the native law would apply ( effectively a form of indirect rule ) until formally superseded by the English law, through Royal Prerogative subjected to the Westminster Parliament.

For example, the cepstrum converts a signal to the frequency domain through Fourier transform, takes the logarithm, then applies another Fourier transform.

The refrigerator rule applies ( you can't stand in a fridge, the door always opens towards you )-If the hinges are on the left then its a left hand ( or left hung ) door.

The same principle applies to more than two segments: given a series of sub-trips at different speeds, if each sub-trip covers the same distance, then the average speed is the harmonic mean of all the sub-trip speeds, and if each sub-trip takes the same amount of time, then the average speed is the arithmetic mean of all the sub-trip speeds.

Assuming that the variance is not infinite and that the central limit theorem applies to the sample then using the delta method, the variance is

For example, when the Supreme Court says that the First Amendment applies in a specific way to suits for slander, then every court is bound by that precedent in its interpretation of the First Amendment as it applies to suits for slander.

In divorce cases, when a court is attempting to distribute marital property, if the divorcing couple is local and the property is local, then the court applies its domestic law lex fori.

One can then argue that since the factual situation is within the British territory, where an American judge applies the English Law, he does not give an extraterritorial application to the foreign rule.

In classical field theory, one writes down a Lagrangian density,, involving a field, φ ( x, t ), and possibly its first derivatives (∂ φ /∂ t and ∇ φ ), and then applies a field-theoretic form of the Euler – Lagrange equation.

From the catch, the rower places the blade in the water, then applies pressure to the oar by simultaneously pushing the seat toward the bow of the boat by extending the legs.

If the error ranges from-1 to + 1, with the analog-to-digital converter used having a resolution of 0. 25, then the input variable's fuzzy set ( which, in this case, also applies to the output variable ) can be described very simply as a table, with the error / delta / output values in the top row and the truth values for each membership function arranged in rows beneath:

The operator dips the brush in plating solution then applies it to the item, moving the brush continually to get an even distribution of the plating material.

It then applies move-to-front transform and Huffman coding.

Maimonides, and other rabbinical commentators, extrapolated this into the conclusion that, if they exist, then male sons and their descendants are the heirs of an individual, but if they do not it would be any daughters or their descendants, and if these do not exist then it would be the individual's father, and if he is no longer alive then the rule concerning heirs applies to him-the father's sons ( the individual's brothers ) and their descendants have priority, followed by the father's daughters ( the individual's sisters ), followed by the father's father ( the individual's grandfather ), and so on.

But if Callicles accepts the counterexample, then he must either withdraw his claim or modify it so that the counterexample no longer applies.

Similarly, first applies to, printing the letter and returning ; this result of is then applied to as in the previous example.

Usually, to evaluate an expression of the form, unlambda first evaluates x, then y, and then applies x to y.

However, if x evaluates to the special value, then y is not evaluated ; instead, the value of the expression is a special " delayed computation " object, which, when applied to an argument z, evaluates y, and then applies its value to z.

* In the plane ( d = 2 ), if there are b vertices on the convex hull, then any triangulation of the points has at most 2n − 2 − b triangles, plus one exterior face ( see Euler characteristic ).

The Euler – MacLaurin summation formula then follows as an integral over the latter.

If the graph G is connected, then the rank of the free group is equal to 1 − χ ( G ): one minus the Euler characteristic of G.

Paul Euler was a friend of the Bernoulli family — Johann Bernoulli, who was then regarded as Europe's foremost mathematician, would eventually be the most important influence on young Leonhard.

If the generalized coordinates are represented as a vector and time differentiation is represented by a dot over the variable, then the equations of motion ( known as the Lagrange or Euler – Lagrange equations ) are a set of equations:

His work is notable for the use of the zeta function ζ ( s ) ( for real values of the argument " s ", as are works of Leonhard Euler, as early as 1737 ) predating Riemann's celebrated memoir of 1859, and he succeeded in proving a slightly weaker form of the asymptotic law, namely, that if the limit of π ( x )/( x / ln ( x )) as x goes to infinity exists at all, then it is necessarily equal to one.

In number theory, Euler's theorem ( also known as the Fermat – Euler theorem or Euler's totient theorem ) states that if n and a are coprime positive integers, then

The Euler – Lagrange equations of motion for the functional E are then given in local coordinates by

So for instance if you have a sphere with a " dent ", then its total curvature is 4π ( the Euler characteristic of a sphere being 2 ), no matter how big or deep the dent.

The geodesic curvature of geodesics being zero, and the Euler characteristic of T being 1, the theorem then states that the sum of the turning angles of the geodesic triangle is equal to 2π minus the total curvature within the triangle.

More generally, if the polyhedron has Euler characteristic ( where g is the genus, meaning " number of holes "), then the sum of the defect is

If ( d < sub > n </ sub >: A < sub > n </ sub > → A < sub > n-1 </ sub >) is a chain complex such that all but finitely many A < sub > n </ sub > are zero, and the others are finitely generated abelian groups ( or finite dimensional vector spaces ), then we can define the Euler characteristic

If the Lagrangian of a system is known, then the equations of motion of the system may be obtained by a direct substitution of the expression for the Lagrangian into the Euler – Lagrange equation.

Let A, B, C denote the vertex angles of the reference triangle, and let x: y: z be a variable point in trilinear coordinates ; then an equation for the Euler line is

Euler noted that if there were only a finite number of primes, then the product on the right would clearly converge, contradicting the divergence of the harmonic series.

* Normalization: If E is a line bundle, then where is the Euler class of the underlying real vector bundle.

The Euler – Lagrange equation, then, is given by

In the two-dimensional realm, Mark Drela and Michael Giles, then graduate students at MIT, developed the ISES Euler program ( actually a suite of programs ) for airfoil design and analysis.

If is a fibration with fiber F, with the base B path-connected, and the fibration is orientable over a field K, then the Euler characteristic with coefficients in the field K satisfies the product property:

An Eulerian orientation of an undirected graph G is an assignment of a direction to each edge of G such that, at each vertex v, the indegree of v equals the outdegree of v. Such an orientation exists for any undirected graph in which every vertex has even degree, and may be found by constructing an Euler tour in each connected component of G and then orienting the edges according to the tour.

Substituting V and E into the Euler relation solved for F, one then obtains

where ƒ is a smooth function, you could use the Euler – Maclaurin formula to convert the series into an integral, plus some corrections involving derivatives of S ( x ), then for large values of a you could use " stationary phase " method to calculate the integral and give an approximate evaluation of the sum.

A quantitative form of Dirichlet's theorem states that if N ≥ 2 is an integer and a is coprime to N, then the proportion of the primes p congruent to a mod N is asymptotic to 1 / n, where n = φ ( N ) is the Euler totient function.