due to the orthogonality relation

Another orthogonality relation is the closure equation:

For the spherical Bessel functions the orthogonality relation is:

Using the orthogonality relation where is the Kronecker delta, we simplify the above three terms for each to see

They satisfy the orthogonality relation

The relation of hyperbolic orthogonality actually applies to classes of parallel lines in the plane, where any particular line can represent the class.

The first orthogonality relation is

The third is true because orthogonality is a symmetric relation.

Subtracting the two, we get the Galerkin orthogonality relation for the error, which is the error between the solution of the original problem,, and the solution of the Galerkin equation,

means that the term in parentheses at left is the transverse part of. Note that this orthogonality relation holds only when X is a timelike unit vector of a Lorenzian Manifold.

) This orthogonality condition can be used to derive the formula for the IDFT from the definition of the DFT, and is equivalent to the unitarity property below.

* Every Jacobi-like polynomial sequence can have its domain shifted and / or scaled so that its interval of orthogonality is, and has Q

* Every Laguerre-like polynomial sequence can have its domain shifted, scaled, and / or reflected so that its interval of orthogonality is, and has Q =

* Every Hermite-like polynomial sequence can have its domain shifted and / or scaled so that its interval of orthogonality is, and has Q

It can be shown that orthogonality of functions is a generalization of the concept of orthogonality of vectors.

The Gauss sum can thus be written as a linear combination of Gaussian periods ( with coefficients χ ( a )); the converse is also true, as a consequence of the orthogonality relations for the group ( Z / nZ )< sup >×</ sup >.

This orthogonality can best be understood in a thought experiment: Consider a model of a population of animals such as crocodiles or tangle web spiders.

So, the direct product of two irreducible representations can also be turned into a unitary representations and now, we have the neat orthogonality property allowing us to decompose the direct product into a direct sum of irreducible representations ( we're also using the property that for finite dimensional representations, if you keep taking proper subrepresentations, you'll hit an irreducible representation eventually.

which can be readily verified using the orthogonality relationship described below.

These theorems can be proven using the orthogonality property.

A shorter, non-numerical example can be found in orthogonality principle.

where the last step can be seen to follow e. g. from a Taylor series expansion, and due to the orthogonality of the states and we have

The resultant throughput reduction can be partly compensated with a large tone set so that each symbol represents several data bits ; a long symbol interval allows these tones to be packed more closely in frequency while maintaining orthogonality.

The image above shows the sign to be negative: to maintain orthogonality, if du < sub > ρ </ sub > is positive with dθ, then du < sub > θ </ sub > must decrease.

If the polynomials f are such that the term on the left is zero, and for, then the orthogonality relationship will hold:

Conversely, if the Doppler spread is large while the delay spread is small, then a shorter symbol period may permit coherent tone detection and the tones must be spaced more widely to maintain orthogonality.

These systems differ in their orthogonality, the Geek Code is very orthogonal in the computer science sense ( may be projected onto basis vectors ), where the Yerkes system is very orthogonal in the taxonomic sense ( represent mutually exclusive classes ).

the interval of orthogonality is, still have two parameters to be determined.

Substituting these expressions into the entropy integral, exchanging the order of integration and summation, and using the orthogonality of the cosines, the entropy may be written:

However, in practice ( as the calculations are performed in floating point arithmetic where inaccuracy is inevitable ), the orthogonality is quickly lost and in some cases the new vector could even be linearly dependent on the set that is already constructed.

There were no specific input or output instructions ; the PDP-11 used memory-mapped I / O and so the same move instruction was used ; orthogonality even enabled moving data directly from an input device to an output device.

The " up tack " symbol ( U + 22A5: ⊥) used by philosophers and logicians ( see contradiction ) also appears, but is often avoided due to its usage for orthogonality.

The lemma is named after Issai Schur who used it to prove Schur orthogonality relations and develop the basics of the representation theory of finite groups.

Reciprocity is also a basic lemma that is used to prove other theorems about electromagnetic systems, such as the symmetry of the impedance matrix and scattering matrix, symmetries of Green's functions for use in boundary-element and transfer-matrix computational methods, as well as orthogonality properties of harmonic modes in waveguide systems ( as an alternative to proving those properties directly from the symmetries of the eigen-operators ).

In order to find the optimal coefficients by the orthogonality principle we solve the equation by inverting and multiplying to get

where · denotes the dot product, see also orthogonality for more information.

These concepts date back to the Ancient Babylonians Egyptians, where orthogonality was a useful concept in civil engineering.

In general, uncorrelatedness is not the same as orthogonality, except in the special case where either X or Y has an expected value of 0.

He demonstrated his principle using vector space geometry based on an “ orthogonality condition ” which only worked in systems where the velocities were defined as a single vector or tensor, and thus, as he wrote at p. 347, was “ impossible to test by means of macroscopic mechanical models ”, and was, as he pointed out, invalid in “ compound systems where several elementary processes take place simultaneously ”.

For α = β = 0, these are called the Legendre polynomials ( for which the interval of orthogonality is and the weight function is simply 1 ):

Because Bessel's equation becomes Hermitian ( self-adjoint ) if it is divided by x, the solutions must satisfy an orthogonality relationship for appropriate boundary conditions.

( Although mutual orthogonality is the only condition, these vectors are usually constructed for ease of decoding, for example columns or rows from Walsh matrices.

The orthogonality of the DFT is now expressed as an orthonormality condition ( which arises in many areas of mathematics as described in root of unity ):

However, there is a certain amount of design philosophy similarity ( e. g., considerable orthogonality and flexible addressing modes ), some assembly language syntax resemblance, as well as opcode mnemonic similarity, but the 6809 is a derivative of the 6800 whereas the 68000 was a totally new design.

The second is an elegant argument using orthogonality and is based upon: Mackiw, G. ( 1995 ).

* Logical independence is geometric orthogonality ;

Closely related and important geometrical concepts are perpendicular lines, meaning lines that form right angles at their point of intersection, and orthogonality, which is the property of forming right angles, usually applied to vectors.

Hence orthogonality of vectors is an extension of the concept of perpendicular vectors into higher-dimensional spaces.

The converse is not true ; having a determinant of + 1 is no guarantee of orthogonality, even with orthogonal columns, as shown by the following counterexample.

Here orthogonality is important not only for reducing A < sup > T </ sup > A = ( R < sup > T </ sup > Q < sup > T </ sup >) QR to R < sup > T </ sup > R, but also for allowing solution without magnifying numerical problems.

The construction of orthogonality of vectors is motivated by a desire to extend the intuitive notion of perpendicular vectors to higher-dimensional spaces.

The extension to either Euclidean or Minkowskian geometry is achieved by adding various further axioms of orthogonality, etc