For the case of a purely inertial autonavigator consisting of three restrained gyros, a coordinate system is used where the sensitive axis of the X accelerometer is parallel to the east-west direction at the base point, and the Y accelerometer sensitive axis is parallel to the north-south direction at the base point.

In this case the relation from X to Y is the subset G of X × Y, and " from X to Y " must always be either specified or implied by the context when referring to the relation.

* Corollary If X is a Banach space, then X is reflexive if and only if X ′ is reflexive, which is the case if and only if its unit ball is compact in the weak topology.

The case where X is F, and we have a bilinear form, is particularly useful ( see for example scalar product, inner product and quadratic form ).

A prime example is in the case of Street Fighter X Tekken, where 12 characters advertised as downloadable content were found completed within the disc.

A frequent particular case occurs when f is a function from X to another set Y ; if x < sub > 1 </ sub > ~ x < sub > 2 </ sub > implies f ( x < sub > 1 </ sub >) = f ( x < sub > 2 </ sub >) then f is said to be a morphism for ~, a class invariant under ~, or simply invariant under ~.

The previous case K < nowiki ></ nowiki > X < nowiki ></ nowiki > is a special case of this.

There, he is confronted by 2 laser-armed trucks, and then the Super X, a piloted VTOL craft constructed in secret to defend Tokyo in case of emergency, in particular a nuclear attack.

In the case of haemophilia, it is sex-linked thus only carried on the X chromosome.

Empirical tests with a 4. 2 GB fragmented file on a DVD media have shown that Microsoft Windows XP supports this, while Mac OS X ( as of 10. 4. 8 ) does not handle this case properly.

In the case of Mac OS X, the driver appears not to support file fragmentation at all ( i. e. it only supports ISO 9660 Level 2 but not Level 3 ).

In this case, the analogy applies both to the form of the words and to their meaning: in each pair, the first word means " one of X ", while the second " two or more of X ", and the difference is always the plural form-s affixed to the second word, signaling the key distinction between singular and plural entities.

An object ( in this case a pencil ) part immersed in water looks bent due to refraction: the light waves from X change direction and so seem to originate at Y.

In the case where each of the X < sub > i </ sub > has a standard normal distribution, the mean range is given by

For the more general case of M non-overlapping populations, X < sub > 1 </ sub > through X < sub > M </ sub >, and the aggregate population:

⟨ H ⟩, be the group generated by H. Then the word problem in H < sup >*</ sup > is solvable: given two words h, k in the generators H of H < sup >*</ sup >, write them as words in X and compare them using the solution to the word problem in G. It is easy to think that this demonstrates a uniform solution the word problem for the class K ( say ) of finitely generated groups that can be embedded in G. If this were the case the non-existence of a universal solvable word problem group would follow easily from Boone-Rogers.

* Racism: Although this was not initially the case, Professor X has come to be compared to civil rights leader Martin Luther King, Jr. and Magneto to the more militant Malcolm X.

But if this is the case, then an arbitrary pencil of lines having a point, P, of **zg as vertex is transformed into a ruled surface of order Af having Af generators concurrent at P.

In the first case the fixed elements within each pencil are the multiple secant and the line joining the vertex, P, to the intersection of **zg and the plane of the pencil which does not lie on the multiple secant.

Thus, if what is at issue is whether `` All S is P '', it is indifferent whether `` Some S is not P '' or `` No S is P '', since in either case the judgment in question is false.

The best presentation of the case for Ambrose is by P. A. Ballerini in his complete edition of that father's works.

* If the balance factor of R is-1, a single left rotation ( with P as the root ) is needed ( Right-Right case ).

The second is a left rotation with P as the root ( Right-Left case ).

* If the balance factor of L is + 1, a single right rotation ( with P as the root ) is needed ( Left-Left case ).

The second is a right rotation with P as the root ( Left-Right case ).

In addition to the balancing described above for insertions, if the balance factor for the tree is 2 and that of the left subtree is 0, a right rotation must be performed on P. The mirror of this case is also necessary.

In the case of parallelization, Amdahl's law states that if P is the proportion of a program that can be made parallel ( i. e., benefit from parallelization ), and ( 1 − P ) is the proportion that cannot be parallelized ( remains serial ), then the maximum speedup that can be achieved by using N processors is

P is thought to be a strict subset in both cases ( and demonstrably cannot be strict in one case but not the other ).

In the nonrelativistic case, electron degeneracy pressure gives rise to an equation of state of the form, where P is the pressure, is the mass density, and is a constant.

catenary is suspended at given points P < sub > 1 </ sub > and P < sub > 2 </ sub > and with given length s. The equation can be determined in this case as follows:

According to Bertrand Russell's Theory of Descriptions, the negation operator in a singular sentence can take either wide or narrow scope: we distinguish between " some S is not P " ( where negation takes " narrow scope ") and " it is not the case that ' some S is P '" ( where negation takes " wide scope ").

Therefore whenever ( G ) appears before another representative lowercase letter ( subject ), the statement would read, " It is always Going to be the case ..." and whenever ( F ) appears before another subject, the statement would read, " At sometime in the Future it will be the case ...." Whenever ( H ) appears before another subject, the statement would read, " It Has always been the case ..." and whenever ( P ) appears before another subject, the statement would read, " At sometime in the Past it has been the case ...."

Geometric arrangement for Fresnel's calculation Consider the case of a point source located at a point P < sub > 0 </ sub >, vibrating at a frequency f. The disturbance may be described by a complex variable U < sub > 0 </ sub > known as the complex amplitude.

In this case, the above rule for Jacobian matrices is usually written as:

terms ), so it will first be given explicitly for the case of 2-by-2 matrices and 3-by-3 matrices, followed by the rule for arbitrary size matrices, which subsumes these two cases.

If G = GL < sub >*</ sub >( K ), then the set of natural numbers is a proper subset of G < sub > 0 </ sub >, since for each natural number n, there is a corresponding identity matrix of dimension n. G ( m, n ) is empty unless m = n, in which case it is the set of all nxn invertible matrices.

In the case where V is of finite dimension n it is common to choose a basis for V and identify GL ( V ) with GL ( n, K ) the group of n-by-n invertible matrices on the field K.

Positive definite matrices are closely related to a positive-definite symmetric bilinear forms ( or sesquilinear forms in the complex case ), and to inner products of vector spaces.

This may be confusing in the special case where or is zero, because we usually don't think of matrices with 0 rows or 0 columns.

( In this case, ' large ' means, approximately, more than 100, 000 elements in each matrix, or enough addressable memory such that the matrices will not fit in L1 and L2 caches.

In the case of rotating frames, the movement composition is simpler than the general case because the final matrix is always a product of rotation matrices.

Common types of matrices include: plates of metal, usually copper or zinc for engraving or etching ; stone, used for lithography ; blocks of wood for woodcuts, linoleum for linocuts and fabric in the case of screen-printing.

The outer product of vectors can be also regarded as a special case of the Kronecker product of matrices.

In the case of 3 × 3 matrices, three such rotations suffice ; and by fixing the sequence we can thus describe all 3 × 3 rotation matrices ( though not uniquely ) in terms of the three angles used, often called Euler angles.

More generally, all four are equal if λ belongs to the center of the ring of entries of the matrix, because in this case λX = Xλ for all matrices X.

Matrix multiplication can be extended to the case of more than two matrices, provided that for each sequential pair, their dimensions match.

In this case, the diagonal matrix Σ is uniquely determined by M ( though the matrices U and V are not ).

In the special but common case in which M is just an m × m square matrix with positive determinant whose entries are plain real numbers, then U, V *, and Σ are m × m matrices of real numbers as well, Σ can be regarded as a scaling matrix, and U and V * can be viewed as rotation matrices.

the matrices satisfy various polynomials such as their minimal polynomials, which form a proper ideal ( because they are not all zero, in which case the result is trivial ); one might call this the characteristic ideal, by analogy with the characteristic polynomial.

However, it is not the case that all normal matrices are either unitary or ( skew -) Hermitian.

The spectral theorem for normal matrices can be seen as a special case of the more general result which holds for all square matrices: Schur decomposition.

( As a special case, the complex numbers may be embedded in the normal real matrices by the mapping, which preserves addition and multiplication.