Given Euler's totient function, and

Given r, the rate of rotation is easy to infer from the constant angular momentum L, so a 2D solution can be easily reconstructed from a 1D solution of this equation.

Given an equilateral triangle, the counterclockwise rotation by 120 ° around the center of the triangle " acts " on the set of vertices of the triangle by mapping every vertex to another one.

Given the irregular shape of the nucleus, Halley's rotation is likely to be complex.

Given that any frame in the space can be described by a rotation matrix, the displacement among them can also be described by a rotation matrix.

Given a unit vector n in R < sup > 3 </ sup > and an angle φ, let R ( φ, n ) represent a counterclockwise rotation about the axis through n ( with orientation determined by n ).

Given the above, for every point in this ball there is a rotation, with axis through the point and the origin, and rotation angle equal to the distance of the point from the origin.

Given a rotation matrix R, a vector u parallel to the rotation axis must satisfy

Given a unit vector u = ( u < sub > x </ sub >, u < sub > y </ sub >, u < sub > z </ sub >), where u < sub > x </ sub >< sup > 2 </ sup > + u < sub > y </ sub >< sup > 2 </ sup > + u < sub > z </ sub >< sup > 2 </ sup > = 1, the matrix for a rotation by an angle of θ about an axis in the direction of u is

Given a triangular orthobicupola, a 60-degree rotation of one cupola before the joining yields a cuboctahedron.

Given a rotation axis z represented by a unit vector k

Given that torques are pseudo-vectors, and hence add linearly, it follows that the rotation speed of the rocket around the aforementioned axis can only increase.

Given x ∈ A, the holomorphic functional calculus allows to define ƒ ( x ) ∈ A for any function ƒ holomorphic in a neighborhood of Furthermore, the spectral mapping theorem holds:

Solomonoff's universal prior probability of any prefix p of a computable sequence x is the sum of the probabilities of all programs ( for a universal computer ) that compute something starting with p. Given some p and any computable but unknown probability distribution from which x is sampled, the universal prior and Bayes ' theorem can be used to predict the yet unseen parts of x in optimal fashion.

Given these definitions, we can state the second part of the Peter – Weyl theorem:

Given an algebra, a homomorphism h thus defines two algebras homomorphic to, the image h () and The two are isomorphic, a result known as the homomorphic image theorem.

Given a source with a fixed voltage and fixed source impedance, the maximum power theorem says this is the only way to extract the maximum power from the source.

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:

Given the Reynolds operator, Hilbert's theorem is proved as follows.

The universal prior probability of any prefix p of a computable sequence x is the sum of the probabilities of all programs ( for a universal computer ) that compute something starting with p. Given some p and any computable but unknown probability distribution from which x is sampled, the universal prior and Bayes ' theorem can be used to predict the yet unseen parts of x in optimal fashion.

Given our current estimate of the parameters θ < sup >( t )</ sup >, the conditional distribution of the Z < sub > i </ sub > is determined by Bayes theorem to be the proportional height of the normal density weighted by τ:

Given the above definitions, Frobenius ' theorem states that a subbundle E is integrable if and only if it arises from a regular foliation of M.

Given two functions, in order to apply the Banach fixed point theorem we want

Given a twice continuously differentiable function f of one real variable, Taylor's theorem for the case n = 1 states that

Given a very ample sheaf L on X and a coherent sheaf F, a theorem of Serre shows that ( the coherent sheaf ) F ⊗ L < sup >⊗ n </ sup > is generated by finitely many global sections for sufficiently large n. This in turn implies that global sections and higher ( Zariski ) cohomology groups

Given a pair, a new index can be constructed using the s < sub > mn </ sub > theorem such that the program coded by ignores its input and merely simulates the computation of the machine with index e on input n.

The invariance theorem follows: Given any description language, our optimal description language is at least as efficient as, with some constant overhead.

Given the RG theorem, the next step in developing a computationally useful method is to determine the fictitious non-interacting system which has the same density as the physical ( interacting ) system of interest.

Given any desired positive integer c, this theorem shows that one can find an algebraic solution approximating a formal power series solution up to the degree specified by c. This leads to theorems that deduce the existence of certain formal moduli spaces of deformations as schemes.

Given a characteristic functional on a nuclear space A, the Bochner – Minlos theorem ( after Salomon Bochner and Robert Adol ' fovich Minlos ) guarantees the existence and uniqueness of the corresponding probability measure on the dual space, given by

Given the small population of the region ( compared with the Central Canadian provinces or the New England states ), the regional economy is a net exporter of natural resources, manufactured goods, and services.

Given the above expression, evidently the result of her ( local ) measurement is that the three-particle state would collapse to one of the following four states ( with equal probability of obtaining each ):

Given that France maintained a strong standing army, and that Prussia and the other German states would need weeks to mobilise their conscript armies, the French held the initial advantage of troop numbers and experience.

Given events, and, Bayes ' rule states that the conditional odds of given are equal to the marginal odds of multiplied by the Bayes factor:

The law states that " given enough eyeballs, all bugs are shallow "; or more formally: " Given a large enough beta-tester and co-developer base, almost every problem will be characterized quickly and the fix will be obvious to someone.

Given an isolated physical system, the allowed states of this system ( i. e. the states the system could occupy without violating the laws of physics ) are part of a Hilbert space H. Some properties of such a space are

Dr. Tennison states: " Given the account of Elliot Paul, and given that Lead Belly witnessed boogie-woogie in 1899 in the Arklatex ; and given the North to South migration of the Thomas family ; and given the Texas & Pacific headquarters in Marshall in the early 1870s ; and given that Harrison County had the largest slave population in the state of Texas ; and given the fact that the best-documented and largest-scale turpentine camps in Texas did not occur until after 1900 in Southeast Texas, it is most probable that boogie-woogie spread from Northeast to Southeast Texas, rather than from Southeast to Northeast Texas, or by having developed diffusely with an even density over all of the Piney Woods of East Texas.

Given the observation space, the state space, a sequence of observations, transition matrix of size such that stores the transition probability of transiting from state to state, emission matrix of size such that stores the probability of observing from state, an array of initial probabilities of size such that stores the probability that. We say a path is a sequence of states that generate the observations.

Linus's law states, " Given enough eyeballs all bugs are shallow.

Given *- representations π, π ' each with unit norm cyclic vectors ξ ∈ H, ξ ' ∈ K such that their respective associated states coincide, then π, π ' are unitarily equivalent representations.

Given an arbitrary ( n ; m ; p ) machine S, such that every two of its states are distinguishable from one another, then there exists an experiment of length

Given two states p and q in S, q simulates p, written p ≤ q if there is a simulation R such that ( p, q ) ∈ R. The relation ≤ is a preorder, and is usually called the simulation preorder.

Given two states and in, is bisimilar to, written, if there is a bisimulation such that is in.

However, the modelling study also states that " Given the complex, three-dimensional nature of the real Pine Island glacier ... it should be clear that the [...] model is a very crude representation of reality.

Given an assignment of every variable to the Marked or Unmarked states, this pa translation reduces to a PA expression, which can be simplified.

Given a particular input, it will always produce the same output, and the underlying machine will always pass through the same sequence of states.

Given that the Republic of Indonesia did not want the RIS parliament to become the legislature of the unitary republic, in May 1950, Hatta and representatives from the federal states agreed to establish a new parliament comprising the 150 members of the RIS parliament, 46 members of the KNIP Working Committee, 13 from the Republic of Indonesia Supreme Advisory council and 32 RIS senators, making 241 members.

Given that the ESCP is a regulation and not a directive, it is arguable whether it has left too many aspects to the discretion of member states, which could call into question the legal certainty expected from a European regulation.

Given that 1 John 2: 1 views Jesus as a paraclete, the reference in John 14: 16 states that Jesus sends a second paraclete to continue the life of the Church after his departure.

Given that the vast majority of criminal sentencing is done at the state level, the American Law Institute and the American Bar Association have each recommended such systems for all the states, and nearly half the states presently have such systems, although significant variations exist among them.